Journal articles: 'Coarse-grid'

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Alber, David M., and Luke N. Olson. "Parallel coarse-grid selection." Numerical Linear Algebra with Applications 14, no. 8 (2007): 611–43. http://dx.doi.org/10.1002/nla.541.

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Evazi, M., and H. Mahani. "Unstructured-Coarse-Grid Generation Using Background-Grid Approach." SPE Journal 15, no. 02 (March 3, 2010): 326–40. http://dx.doi.org/10.2118/120170-pa.

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Summary Reservoir flow simulation involves subdivision of the physical domain into a number of gridblocks. This is best accomplished with optimized gridpoint density and a minimized number of gridblocks, especially for coarse-grid generation from a fine-grid geological model. In any coarse-grid generation, proper distribution of gridpoints, which form the basis of numerical gridblocks, is a challenging task. We show that this can be achieved effectively by a novel grid-generation approach based on a background grid that stores gridpoint spacing parameters. Spacing parameter (L) can be described by Poisson's equation (∇2L = G), where the local density of gridpoints is controlled by a variable source term (G); see Eq. 1. This source term can be based on different gridpoint density indicators, such as permeability variations, fluid velocity, or their combination (e.g., vorticity) where they can be extracted from the reference fine grid. Once a background grid is generated, advancing-front triangulation (AFT) and then Delaunay tessellation are invoked to form the final (coarse) gridblocks. The algorithm produces grids varying smoothly from high- to low-density gridpoints, thus minimizing use of grid-smoothing and -optimization techniques. This algorithm is quite flexible, allowing choice of the gridding indicator, hence providing the possibility of comparing the grids generated with different indicators and selecting the best. In this paper, the capabilities of approach in generation of unstructured coarse grids from fine geological models are illustrated using 2D highly heterogeneous test cases. Flexibility of algorithm to gridding indicator is demonstrated using vorticity, permeability variation, and velocity. Quality of the coarse grids is evaluated by comparing their two-phase-flow simulation results to those of fine grid and uniform coarse grid. Results demonstrate the robustness and attractiveness of the approach, as well as relative quality/performance of grids generated by using different indicators.

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Kehl, René, and Reinhard Nabben. "Avoiding singular coarse grid systems." Linear Algebra and its Applications 507 (October 2016): 137–52. http://dx.doi.org/10.1016/j.laa.2016.05.025.

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Cho, Yongchae, and Richard L. Gibson, Jr. "Reverse time migration via frequency-adaptive multiscale spatial grids." GEOPHYSICS 84, no. 2 (March 1, 2019): S41—S55. http://dx.doi.org/10.1190/geo2018-0292.1.

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Reverse time migration (RTM) is widely used because of its ability to recover complex geologic structures. However, RTM also has a drawback in that it requires significant computational cost. In RTM, wave modeling accounts for the largest part of the computing cost for calculating forward- and backward-propagated wavefields before applying an imaging condition. For this reason, we have applied a frequency-adaptive multiscale spatial grid to enhance the efficiency of the wave simulations. To implement wave modeling for different values of the spatial grid interval, we apply a model reduction technique, the generalized multiscale finite-element method (GMsFEM), which solves local spectral problems on a fine grid to simulate wave propagation on a coarser grid. We can enhance the speed of computation without sacrificing accuracy by using coarser grids for lower frequency waves, while applying a finer grid for higher frequency waves. In the proposed method, we can control the size of the coarse grid and level of heterogeneity of the wave solutions to tune the trade-off between speedup and accuracy. As we increase the expected level of complexity of the wave solutions, the GMsFEM wave modeling can capture more detailed features of waves. After computing the forward and backward wavefield on the coarse grid, we reproject the coarse wave solutions to the fine grid to construct the RTM gradient image. Although wave solutions are computed on a coarse grid, we still obtain the RTM images without reducing the image resolution by projecting coarse wave solutions to the fine grid. We determine the efficiency of the proposed imaging method using the Marmousi-2 model. We compare the RTM images using GMsFEM with a fixed coarse mesh and a multiple frequency-adaptive coarse meshes to indicate the image quality and computational speed of the new approach.

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MacLachlan, S., and Yousef Saad. "A Greedy Strategy for Coarse-Grid Selection." SIAM Journal on Scientific Computing 29, no. 5 (January 2007): 1825–53. http://dx.doi.org/10.1137/060654062.

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Petrovskaya, Natalia B., and Sergei V. Petrovskii. "The coarse-grid problem in ecological monitoring." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2122 (April 29, 2010): 2933–53. http://dx.doi.org/10.1098/rspa.2010.0023.

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Obtaining information about pest-insect population size is an important problem of pest monitoring and control. Usually, this problem has to be solved based on scarce spatial data about the population density. The problem of monitoring can thus be linked to a more general mathematical problem of numerical integration on a coarse grid. Numerical integration on coarse grids has rarely been considered in literature as it is usually assumed that the grid can be refined. However, this is not the case in ecological monitoring where fine grids are not available. In this paper, we introduce a method of numerical integration that allows one to accurately evaluate an integral on a coarse grid. The method is tested on several functions with different properties to show its effectiveness. We then use the method to obtain an estimate of the population size for different population distributions and show that an ecologically reasonable accuracy can be achieved on a very coarse grid consisting of just a few points. Finally, we summarize our mathematical findings as a protocol of ecological monitoring, thus sending a clear and practically important message to ecologists and pest-control specialists.

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Bender, Iring, Roger Horsley, and Werner Wetzel. "Coarse grid Yukawa interaction for staggered fermions." Nuclear Physics B 349, no. 1 (January 1991): 277–304. http://dx.doi.org/10.1016/0550-3213(91)90198-7.

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Krogstad, S., V. L. L. Hauge, and A. F. F. Gulbransen. "Adjoint Multiscale Mixed Finite Elements." SPE Journal 16, no. 01 (August 23, 2010): 162–71. http://dx.doi.org/10.2118/119112-pa.

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Summary We develop an adjoint model for a simulator consisting of a multiscale pressure solver and a saturation solver that works on flow-adapted grids. The multiscale method solves the pressure on a coarse grid that is close to uniform in index space and incorporates fine-grid effects through numerically computed basis functions. The transport solver works on a coarse grid adapted by a fine-grid velocity field obtained by the multiscale solver. Both the multiscale solver for pressure and the flow-based coarsening approach for transport have shown earlier the ability to produce accurate results for a high degree of coarsening. We present results for a complex realistic model to demonstrate that control settings based on optimization of our multiscale flow-based model closely match or even outperform those found by using a fine-grid model. For additional speed-up, we develop mappings used for rapid system updates during the timestepping procedure. As a result, no fine-grid quantities are required during simulations and all fine-grid computations (multiscale basis functions, generation of coarse transport grid, and coarse mappings) become a preprocessing step. The combined methodology enables optimization of waterflooding on a complex model with 45,000 grid cells in a few minutes.

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Ziv, Alexander, and Elena Solov’eva. "Approximate noise maps as instrument for evaluation of the city environment quality." Noise Mapping 8, no. 1 (January 1, 2021): 260–67. http://dx.doi.org/10.1515/noise-2021-0021.

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Abstract The paper discusses noise mapping from the prospective of general evaluation of the state of the city environment. Suggested is a noise evaluation procedure based on a two-step spatial discretization - coarse and fine grids. The coarse grid is used for evaluation of average noise levels (background noise). For this, rather simple method is proposed, where average noise levels are estimated directly for the whole coarse grid cells instead of averaging the noise levels computed point-wise. The fine grid is used for finding the obstacle density to apply in calculations over the coarse grid. It may be used also for additional noise levels detailing in the close vicinity of noise sources where noise propagation is strongly affected by surrounding structures. The detailed results allow correction of the averages over the coarse grid. In comparison with other approaches, the suggested procedure takes little computing time to execute for the entire city. Test example shows reasonable agreement with results computed using the ‘Ecolog-Noise’ software package that has gained popularity in Russian Federation since its introduction in 2008. Another example describes the application of the proposed method for a moderate size densely built city.

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Gavrilieva, Uygulana, Maria Vasilyeva, and Eric T. Chung. "Generalized Multiscale Finite Element Method for Elastic Wave Propagation in the Frequency Domain." Computation 8, no. 3 (July 7, 2020): 63. http://dx.doi.org/10.3390/computation8030063.

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In this work, we consider elastic wave propagation in fractured media. The mathematical model is described by the Helmholtz problem related to wave propagation with specific interface conditions (Linear Slip Model, LSM) on the fracture in the frequency domain. For the numerical solution, we construct a fine grid that resolves all fracture interfaces on the grid level and construct approximation using a finite element method. We use a discontinuous Galerkin method for the approximation by space that helps to weakly impose interface conditions on fractures. Such approximation leads to a large system of equations and is computationally expensive. In this work, we construct a coarse grid approximation for an effective solution using the Generalized Multiscale Finite Element Method (GMsFEM). We construct and compare two types of the multiscale methods—Continuous Galerkin Generalized Multiscale Finite Element Method (CG-GMsFEM) and Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM). Multiscale basis functions are constructed by solving local spectral problems in each local domains to extract dominant modes of the local solution. In CG-GMsFEM, we construct continuous multiscale basis functions that are defined in the local domains associated with the coarse grid node and contain four coarse grid cells for the structured quadratic coarse grid. The multiscale basis functions in DG-GMsFEM are discontinuous and defined in each coarse grid cell. The results of the numerical solution for the two-dimensional Helmholtz equation are presented for CG-GMsFEM and DG-GMsFEM for different numbers of multiscale basis functions.

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Chen, Chuanjun, Wei Liu, and Xin Zhao. "A Two-Grid Finite Element Method for a Second-Order Nonlinear Hyperbolic Equation." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/803615.

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We present a two-grid finite element scheme for the approximation of a second-order nonlinear hyperbolic equation in two space dimensions. In the two-grid scheme, the full nonlinear problem is solved only on a coarse grid of sizeH. The nonlinearities are expanded about the coarse grid solution on the fine gird of sizeh. The resulting linear system is solved on the fine grid. Some a priori error estimates are derived with theH1-normO(h+H2)for the two-grid finite element method. Compared with the standard finite element method, the two-grid method achieves asymptotically same order as long as the mesh sizes satisfyh=O(H2).

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Li, Hangyu, Jeroen C. Vink, and Faruk O. Alpak. "A Dual-Grid Method for the Upscaling of Solid-Based Thermal Reactive Flow, With Application to the In-Situ Conversion Process." SPE Journal 21, no. 06 (June 2, 2016): 2097–111. http://dx.doi.org/10.2118/173248-pa.

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Summary Thermal-reactive compositional-flow simulation in porous media is essential to model thermal-oil-recovery processes for extraheavy-hydrocarbon resources, and an example is the in-situ conversion process (ICP) developed by Shell for oil-shale production. Computational costs can be very high for such a complex system, which makes direct fine-scale simulations prohibitively time-consuming for large field-scale applications. This motivates the use of coarse grids for thermal-reactive compositional-flow simulation. However, significant errors are introduced by use of coarse-scale models without carefully computing the appropriate coarse parameters. In this paper, we develop an innovative dual-grid method to effectively capture the fine-scale reaction rates in coarse-scale ICP-simulation models. In our dual-grid method, coupled thermal-reactive compositional-flow equations are solved only on the coarse scale, with the kinetic parameters (frequency factors) calculated on the basis of fine-scale computations, such as temperature downscaling and fine-scale reaction-rate calculation. A dual-grid treatment for the heater-well model is also developed with coarse-scale heater-well indices calculated on the basis of fine-scale well results. The dual-grid heater-well treatment is able to provide accurate heater temperatures. The newly developed dual-grid method is applied to realistic cross-sectional ICP-pattern models with a vertical production well and multiple horizontal heater wells operated subject to fixed and time-varying heater powers. It is shown that the dual-grid model delivers results that are in close agreement with the fine-scale reference results for all quantities of interest. Despite the fact that the dual-grid method is implemented at the simulation-deck level, by use of the flexible scripting and monitor functionalities of our proprietary simulation package, significant computational improvements are achieved for all cases considered.

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Wu, Chunxiao, and Justin W. L. Wan. "Multigrid Methods with Newton-Gauss-Seidel Smoothing and Constraint Preserving Interpolation for Obstacle Problems." Numerical Mathematics: Theory, Methods and Applications 8, no. 2 (May 2015): 199–219. http://dx.doi.org/10.4208/nmtma.2015.w08si.

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AbstractIn this paper, we propose a multigrid algorithm based on the full approximate scheme for solving the membrane constrained obstacle problems and the minimal surface obstacle problems in the formulations of HJB equations. A Newton-Gauss-Seidel (NGS) method is used as smoother. A Galerkin coarse grid operator is proposed for the membrane constrained obstacle problem. Comparing with standard FAS with the direct discretization coarse grid operator, the FAS with the proposed operator converges faster. A special prolongation operator is used to interpolate functions accurately from the coarse grid to the fine grid at the boundary between the active and inactive sets. We will demonstrate the fast convergence of the proposed multigrid method for solving two model obstacle problems and compare the results with other multigrid methods.

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Li, Jun, and Donald Brown. "Upscaled Lattice Boltzmann Method for Simulations of Flows in Heterogeneous Porous Media." Geofluids 2017 (2017): 1–12. http://dx.doi.org/10.1155/2017/1740693.

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An upscaled Lattice Boltzmann Method (LBM) for flow simulations in heterogeneous porous media at the Darcy scale is proposed in this paper. In the Darcy-scale simulations, the Shan-Chen force model is used to simplify the algorithm. The proposed upscaled LBM uses coarser grids to represent the average effects of the fine-grid simulations. In the upscaled LBM, each coarse grid represents a subdomain of the fine-grid discretization and the effective permeability with the reduced-order models is proposed as we coarsen the grid. The effective permeability is computed using solutions of local problems (e.g., by performing local LBM simulations on the fine grids using the original permeability distribution) and used on the coarse grids in the upscaled simulations. The upscaled LBM that can reduce the computational cost of existing LBM and transfer the information between different scales is implemented. The results of coarse-grid, reduced-order, simulations agree very well with averaged results obtained using a fine grid.

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Vuik, C., and J. Frank. "Coarse grid acceleration of a parallel block preconditioner." Future Generation Computer Systems 17, no. 8 (June 2001): 933–40. http://dx.doi.org/10.1016/s0167-739x(01)00035-8.

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Tufo, H. M., and P. F. Fischer. "Fast Parallel Direct Solvers for Coarse Grid Problems." Journal of Parallel and Distributed Computing 61, no. 2 (February 2001): 151–77. http://dx.doi.org/10.1006/jpdc.2000.1676.

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Dai, Ruxin, Jun Zhang, and Yin Wang. "Multiple coarse grid acceleration for multiscale multigrid computation." Journal of Computational and Applied Mathematics 269 (October 2014): 75–85. http://dx.doi.org/10.1016/j.cam.2014.03.021.

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Gurieva, Y. L., and V. P. Il’in. "On Coarse Grid Correction Methods in Krylov Subspaces." Journal of Mathematical Sciences 232, no. 6 (June 23, 2018): 774–82. http://dx.doi.org/10.1007/s10958-018-3907-9.

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Tyrylgin, Aleksei, Maria Vasilyeva, Dmitry Ammosov, Eric T. Chung, and Yalchin Efendiev. "Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media." Fluids 6, no. 8 (August 23, 2021): 298. http://dx.doi.org/10.3390/fluids6080298.

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In this paper, we consider the poroelasticity problem in fractured and heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements in heterogeneous media. Due to scale disparity, many approaches have been developed for solving detailed fine-grid problems on a coarse grid. However, some approaches can lack good accuracy on a coarse grid and some corrections for coarse-grid solutions are needed. In this paper, we present a coarse-grid approximation based on the generalized multiscale finite element method (GMsFEM). We present the construction of the offline and online multiscale basis functions. The offline multiscale basis functions are precomputed for the given heterogeneity and fracture network geometry, where for the construction, we solve a local spectral problem and use the dominant eigenvectors (appropriately defined) to construct multiscale basis functions. To construct the online basis functions, we use current information about the local residual and solve coupled poroelasticity problems in local domains. The online basis functions are used to enrich the offline multiscale space and rapidly reduce the error using residual information. Only with appropriate offline coarse-grid spaces can one guarantee a fast convergence of online methods. We present numerical results for poroelasticity problems in fractured and heterogeneous media. We investigate the influence of the number of offline and online basis functions on the relative errors between the multiscale solution and the reference (fine-scale) solution.

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Chen, Chuanjun, and Wei Liu. "A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations." Abstract and Applied Analysis 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/391918.

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A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid sizeH. The nonlinearities are expanded about the coarse grid solution on a fine gird of sizeh, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with theH1-normO(h+H2)which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfyh=O(H2). An example is also given to illustrate the theoretical results.

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Si, Weijian, Fuhong Zeng, Changbo Hou, and Zhanli Peng. "A Sparse-Based Off-Grid DOA Estimation Method for Coprime Arrays." Sensors 18, no. 9 (September 10, 2018): 3025. http://dx.doi.org/10.3390/s18093025.

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Recently, many sparse-based direction-of-arrival (DOA) estimation methods for coprime arrays have become popular for their excellent detection performance. However, these methods often suffer from grid mismatch problem due to the discretization of the potential angle space, which will cause DOA estimation performance degradation when the target is off-grid. To this end, we proposed a sparse-based off-grid DOA estimation method for coprime arrays in this paper, which includes two parts: coarse estimation process and fine estimation process. In the coarse estimation process, the grid points closest to the true DOAs, named coarse DOAs, are derived by solving an optimization problem, which is constructed according to the statistical property of the vectorized covariance matrix estimation error. Meanwhile, we eliminate the unknown noise variance effectively through a linear transformation. Due to finite snapshots effect, some undesirable correlation terms between signal and noise vectors exist in the sample covariance matrix. In the fine estimation process, we therefore remove the undesirable correlation terms from the sample covariance matrix first, and then utilize a two-step iterative method to update the grid biases. Combining the coarse DOAs with the grid biases, the final DOAs can be obtained. In the end, simulation results verify the effectiveness of the proposed method.

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CHUNG, ERIC T., YALCHIN EFENDIEV, and RICHARD L. GIBSON. "AN ENERGY-CONSERVING DISCONTINUOUS MULTISCALE FINITE ELEMENT METHOD FOR THE WAVE EQUATION IN HETEROGENEOUS MEDIA." Advances in Adaptive Data Analysis 03, no. 01n02 (April 2011): 251–68. http://dx.doi.org/10.1142/s1793536911000842.

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Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

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Karimi-Fard, M., and L. J. J. Durlofsky. "Accurate Resolution of Near-Well Effects in Upscaled Models Using Flow-Based Unstructured Local Grid Refinement." SPE Journal 17, no. 04 (November 29, 2012): 1084–95. http://dx.doi.org/10.2118/141675-pa.

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Summary We present a new approach for representing wells in coarse-scale reservoir simulation models. The technique is based on an expanded well model concept which provides a systematic procedure for the construction of the near-well grid. The method proceeds by first defining an underlying fine-scale model, in which the well and any key near-well features such as hydraulic fractures are fully resolved using an unstructured grid. In the (coarse) simulation model, the geometry of the grid in the expanded well region, and the associated "radial" transmissibilities, are determined from the solution of a fine-scale, single-phase, well-driven flow problem. The coarse-scale transmissibilities outside of the well region are computed using existing local upscaling techniques or by applying a new global upscaling procedure. Thus, through use of near-well flow-based gridding and generalized local grid refinement, this methodology efficiently incorporates the advantages of highly-resolved unstructured grid representations of wells into coarse models. The overall model provided by this technique is compatible with any reservoir simulator that allows general unstructured cell-to-cell connections (model capabilities, in terms of flow physics, are defined by the simulator). The expanded well modeling approach is applied to challenging 3D problems involving injection and production in a low-permeability heterogeneous reservoir, tight-gas production by a hydraulically-fractured well, and production in a gas-condensate reservoir. In the first two cases, where it is possible to simulate the fine-grid unstructured model, results using the expanded well model closely match the reference solutions, while standard approaches lead to significant error. In the gas-condensate example, which involves a nine-component compositional model, the reference solution is not computed, but the solution using the expanded well model is shown to be physically reasonable while standard coarse-grid solutions show large variation under grid refinement.

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Correia, Manuel Gomes, Célio Maschio, and Denis José Schiozer. "Flow Simulation Using Local Grid Refinements to Model Laminated Reservoirs." Oil & Gas Sciences and Technology – Revue d’IFP Energies nouvelles 73 (2018): 5. http://dx.doi.org/10.2516/ogst/2017043.

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Super-giant carbonate fields, such as Ghawar, in Saudi Arabia, and Lula, at the Brazilian pre-salt, show highly heterogeneous behavior that is linked to high permeability intervals in thin layers. This article applies Local Grid Refinements (LGR) integrated with upscaling procedures to improve the representation of highly laminated reservoirs in flow simulation by preserving the static properties and dynamic trends from geological model. This work was developed in five main steps: (1) define a conventional coarse grid, (2) define LGR in the conventional coarse grid according to super-k and well locations, (3) apply an upscaling procedure for all scenarios, (4) define LGR directly in the simulation model, without integrate geological trends in LGR and (5) compare the dynamic response for all cases. To check results and compare upscaling matches, was used the benchmark model UNISIM-II-R, a refined model based on a combination of Brazilian Pre-salt and Ghawar field information. The main results show that the upscaling of geological models for coarse grid with LGR in highly permeable thin layers provides a close dynamic representation of geological characterization compared to conventional coarse grid and LGR only near-wells. Pseudo-relative permeability curves should be considered for (a) conventional coarse grid or (b) LGR scenarios under dual-medium flow simulations as the upscaling of discrete fracture networks and dual-medium flow models presents several limitations. The conventional approach of LGR directly in simulation model, presents worse results than LGR integrated with upscaling procedures as the extrapolation of dynamic properties to the coarse block mismatch the dynamic behavior from geological characterization. This work suggests further improvements for results for upscaling procedures that mask the flow behavior in highly laminated reservoirs.

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Ferm, Lars. "The Number of Coarse-Grid Iterations Every Cycle for the Two-Grid Method." SIAM Journal on Scientific Computing 19, no. 2 (March 1998): 493–501. http://dx.doi.org/10.1137/s1064827592234314.

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Tsanis, Ioannis K., and Jian Wu. "A Nested-Grid Hydrodynamic/Pollutant Transport Model for Nearshore Areas in Hamilton Harbour." Water Quality Research Journal 30, no. 2 (May 1, 1995): 205–30. http://dx.doi.org/10.2166/wqrj.1995.022.

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Abstract A nested-grid depth-averaged circulation model was developed and applied to three nearshore areas in Hamilton Harbour: the western basin, LaSalle Park waterfront and the northeastern shoreline. The grid sizes used were 100 m for the whole harbour, and 25 m for the three nearshore areas. General features of current circulation and horizontal mixing times under various wind directions and speeds were obtained for the whole harbour using the coarse-grid model. The fine-grid model (water elevations and current information on the open boundaries were obtained from the whole harbour model) then provided current patterns which were used to drive the pollutant transport model. Simulation results reveal that the current in the fine-grid model is close to the current from the coarse-grid model, while more detailed current structures are explored. The water elevations from the fine-grid model agree well with the elevations from the coarse-grid one. The impact of artificial islands was examined by studying changes in current patterns, pollutant peaks, exposure and flushing time in different locations of concern. The design proposed provides: (i) minimum change in the existing current patterns; (ii) avoidance of pollutant hot spots; and (iii) minimum changes in the flushing time of pollutants.

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Iliev, O., R. Lazarov, and J. Willems. "Numerical Study of Two-grid Preconditioners for 1-d Elliptic Problems with Highly Oscillating Discontinuous Coefficients." Computational Methods in Applied Mathematics 7, no. 1 (2007): 48–67. http://dx.doi.org/10.2478/cmam-2007-0003.

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AbstractVarious advanced two-level iterative methods are studied numerically and compared with each other in conjunction with finite volume discretizations of symmetric 1-D elliptic problems with highly oscillatory discontinuous coefficients. Some of the methods considered rely on the homogenization approach for deriving the coarse grid operator. This approach is considered here as an alternative to the well-known Galerkin approach for deriving coarse grid operators. Different intergrid transfer operators are studied, primary consideration being given to the use of the so-called problemdependent prolongation. The two-grid methods considered are used as both solvers and preconditioners for the Conjugate Gradient method. The recent approaches, such as the hybrid domain decomposition method introduced by Vassilevski and the globallocal iterative procedure proposed by Durlofsky et al. are also discussed. A two-level method converging in one iteration in the case where the right-hand side is only a function of the coarse variable is introduced and discussed. Such a fast convergence for problems with discontinuous coefficients arbitrarily varying on the fine scale is achieved by a problem-dependent selection of the coarse grid combined with problem-dependent prolongation on a dual grid. The results of the numerical experiments are presented to illustrate the performance of the studied approaches.

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Yang, Ying. "Terrain Real-Time Rendering Based on Projected Grid and GPU Unsaturated Error Metric." Advanced Materials Research 926-930 (May 2014): 3281–85. http://dx.doi.org/10.4028/www.scientific.net/amr.926-930.3281.

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To render terrain realistically and efficiently, we firstly introduce projected grid to construct terrain coarse meshes. Projected grid uses longest edge bisection, taking advantages of fracture eliminating, to implement triangle subdivision. To realize adaptive refinement of terrain coarse meshes, we introduce the concept of bounding sphere and the idea of unsaturated error metric, design GPU unsaturated error metric based on bounding sphere, in order to implement terrain LOD modeling finally. Experimental results show that projected grid avoids culling and improves the speed of terrain rendering, GPU unsaturated error metric can select terrain coarse meshes nodes accurately, reduce triangles number and keep terrain feature. As a result, the algorithm can achieve a high FPS and a good visual effect, implement terrain real-time rendering.

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Li, Zong Zhe, Zheng Hua Wang, Lu Yao, and Wei Cao. "A Combined Global Coarsening Method for 3D Multigrid Applications." Applied Mechanics and Materials 236-237 (November 2012): 1049–53. http://dx.doi.org/10.4028/www.scientific.net/amm.236-237.1049.

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An automatic agglomeration methodology to generate coarse grids for 3D flow solutions on anisotropic unstructured grids has been introduced in this paper. The algorithm combines isotropic octree based coarsening and anisotropic directional agglomeration to yield a desired coarsening ratio and high quality of coarse grids, which developed for cell-centered multigrid applications. This coarsening strategy developed is presented on an unstructured grid over 3D ONERA M6 wing. It is shown that the present method provides suitable coarsening ratio and well defined aspect ratio cells at all coarse grid levels.

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Lashuk, Ilya V., and Panayot S. Vassilevski. "The Construction of the Coarse de Rham Complexes with Improved Approximation Properties." Computational Methods in Applied Mathematics 14, no. 2 (April 1, 2014): 257–303. http://dx.doi.org/10.1515/cmam-2014-0004.

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Abstract. We present two novel coarse spaces (H1- and $H(\operatorname{curl})$-conforming) based on element agglomeration on unstructured tetrahedral meshes. Each H1-conforming coarse basis function is continuous and piecewise-linear with respect to an original tetrahedral mesh. The $H(\operatorname{curl})$-conforming coarse space is a subspace of the lowest order Nédélec space of the first type. The H1-conforming coarse space exactly interpolates affine functions on each agglomerate. The $H(\operatorname{curl})$-conforming coarse space exactly interpolates vector constants on each agglomerate. Combined with the $H(\operatorname{div})$- and L2-conforming spaces developed previously in [Numer. Linear Algebra Appl. 19 (2012), 414–426], the newly constructed coarse spaces form a sequence (with respect to exterior derivatives) which is exact as long as the underlying sequence of fine-grid spaces is exact. The constructed coarse spaces inherit the approximation and stability properties of the underlying fine-grid spaces supported by our numerical experiments. The new coarse spaces, in addition to multigrid, can be used for upscaling of broad range of PDEs involving $\operatorname{curl}$, $\operatorname{div}$ and $\operatorname{grad}$ differential operators.

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Du, Shouhong, Larry S. Fung, and Ali H. Dogru. "Aquifer Acceleration in Parallel Implicit Field-Scale Reservoir Simulation." SPE Journal 23, no. 02 (February 12, 2018): 614–24. http://dx.doi.org/10.2118/182686-pa.

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Summary Grid coarsening outside of the areas of interest is a common method to reduce computational cost in reservoir simulation. Aquifer regions are candidates for grid coarsening. In this situation, upscaling is applied to the fine grid to generate coarse-grid flow properties. The efficacy of the approach can be judged easily by comparing the simulation results between the coarse-grid model and the fine-grid model. For many reservoirs in the Middle East bordered by active aquifers, transient water influx is an important recovery mechanism that needs to be modeled correctly. Our experience has shown that the standard grid coarsening and upscaling method do not produce correct results in this situation. Therefore, the objective of this work is to build a method that retains the fine-scale heterogeneities to accurately represent the water movement, but to significantly reduce the computational cost of the aquifer grids in the model. The new method can be viewed as a modified two-level multigrid (MTL-MG) or a specialized adaptation of the multiscale method. It makes use of the vertical-equilibrium (VE) concept in the fine-scale pressure reconstruction in which it is applicable. The method differs from the standard grid coarsening and upscaling method in which the coarse-grid properties are computed a priori. Instead, the fine-scale information is restricted to the coarse grid during Newton's iteration to represent the fine-scale flow behavior. Within the aquifer regions, each column of fine cells is coarsened vertically based on fine-scale z-transmissibility. A coarsened column may consist of a single amalgamated aquifer cell or multiple vertically disconnected aquifer cells separated by flow barriers. The pore volume (PV), compressibility, and lateral flow terms of the coarse cell are restricted from the fine-grid cells. The lateral connectivity within the aquifer regions and the one between the aquifer and the reservoir are honored, inclusive of the fine-scale description of faults, pinchouts, and null cells. Reservoir regions are not coarsened. Two alternatives exist for the fine-scale pressure reconstruction from the coarse-grid solution. The first method uses the VE concept. When VE applies, pressure variation can be analytically computed in the solution update step. Otherwise, the second method is to apply a 1D z-line solve for the fine-scale aquifer pressure from the coarse-grid solution. Simulation results for several examples are included to demonstrate the efficacy and efficiency of the method. We have applied the method to several Saudi Arabian complex full-field simulation models in which the transient aquifer water influx has been identified as a key factor. These models include dual-porosity/dual-permeability (DPDP) models, as well as models with faults and pinchouts in corner-point-geometry grids, for both history match and prediction period. The method is flexible and allows for the optional selection of aquifer regions to be coarsened, either only peripheral aquifers or both the peripheral and bottom aquifers. The new method gives nearly identical results compared with the original runs without coarsening, but with significant reduction in computer time or hardware cost. These results will be detailed in the paper.

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Harris, Lucas M., and Dale R. Durran. "An Idealized Comparison of One-Way and Two-Way Grid Nesting." Monthly Weather Review 138, no. 6 (June 1, 2010): 2174–87. http://dx.doi.org/10.1175/2010mwr3080.1.

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Abstract Most mesoscale models can be run with either one-way (parasitic) or two-way (interactive) grid nesting. This paper presents results from a linear 1D shallow-water model to determine whether the choice of nesting method can have a significant impact on the solution. Two-way nesting was found to be generally superior to one-way nesting. The only situation in which one-way nesting performs better than two-way is when very poorly resolved waves strike the nest boundary. A simple filter is proposed for use exclusively on the coarse-grid values within the sponge zone of an otherwise conventional sponge boundary condition (BC). The two-way filtered sponge BC gives better results than any of the other methods considered in these tests. Results for all wavelengths were found to be robust to other changes in the formulation of the sponge boundary, particularly with the width of the sponge layer. The increased reflection for longer-wavelength disturbances in the one-way case is due to a phase difference between the coarse- and nested-grid solutions at the nested-grid boundary that accumulates because of the difference in numerical phase speeds between the grids. Reflections for two-way nesting may be estimated from the difference in numerical group velocities between the coarse and nested grids, which only becomes large for waves that are poorly resolved on the coarse grid.

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Jenssen, Carl B., and Per A. Weinerfelt. "Coarse grid correction scheme for implicit multiblock Euler calculations." AIAA Journal 33, no. 10 (October 1995): 1816–21. http://dx.doi.org/10.2514/3.12732.

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Dubois, Olivier, Martin J. Gander, Sébastien Loisel, Amik St-Cyr, and Daniel B. Szyld. "The Optimized Schwarz Method with a Coarse Grid Correction." SIAM Journal on Scientific Computing 34, no. 1 (January 2012): A421—A458. http://dx.doi.org/10.1137/090774434.

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YAMADA, Tomonori. "Mesh Free Analysis by Selective Structured Coarse Grid Correction." Proceedings of The Computational Mechanics Conference 2004.17 (2004): 463–64. http://dx.doi.org/10.1299/jsmecmd.2004.17.463.

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Viellieber, Mathias, Philipp Dietrich, and Andreas G. Class. "Coarse-Grid-CFD for a Wire Wrapped Fuel Assembly." PAMM 13, no. 1 (November 29, 2013): 317–18. http://dx.doi.org/10.1002/pamm.201310154.

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Wang, Chien-Yen, Chan-Yun Yang, Shadi Banitaan, Chaomin Luo, and Sainzaya Galsanbadam. "Coarse grid partition to speed up A* robot navigation." Journal of the Chinese Institute of Engineers 43, no. 2 (December 5, 2019): 186–99. http://dx.doi.org/10.1080/02533839.2019.1694444.

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Yavneh, Irad. "Coarse-Grid Correction for Nonelliptic and Singular Perturbation Problems." SIAM Journal on Scientific Computing 19, no. 5 (September 1998): 1682–99. http://dx.doi.org/10.1137/s1064827596310998.

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Wu, Shu-Lin. "Toward Parallel Coarse Grid Correction for the Parareal Algorithm." SIAM Journal on Scientific Computing 40, no. 3 (January 2018): A1446—A1472. http://dx.doi.org/10.1137/17m1141102.

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Schumann, Guy J. P., Konstantinos M. Andreadis, and Paul D. Bates. "Downscaling coarse grid hydrodynamic model simulations over large domains." Journal of Hydrology 508 (January 2014): 289–98. http://dx.doi.org/10.1016/j.jhydrol.2013.08.051.

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Notay, Yvan, and Artem Napov. "Further comparison of additive and multiplicative coarse grid correction." Applied Numerical Mathematics 65 (March 2013): 53–62. http://dx.doi.org/10.1016/j.apnum.2012.12.001.

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Gibson, Richard L., Kai Gao, Eric Chung, and Yalchin Efendiev. "Multiscale modeling of acoustic wave propagation in 2D media." GEOPHYSICS 79, no. 2 (March 1, 2014): T61—T75. http://dx.doi.org/10.1190/geo2012-0208.1.

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Conventional finite-difference methods produce accurate solutions to the acoustic and elastic wave equation for many applications, but they face significant challenges when material properties vary significantly over distances less than the grid size. This challenge is likely to occur in reservoir characterization studies, because important reservoir heterogeneity can be present on scales of several meters to ten meters. Here, we describe a new multiscale finite-element method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine- and coarse-scale grids. The wave equation is solved on a coarse grid, but it uses basis functions that are generated from the fine grid and allow the representation of the fine-scale variation of the wavefield on the coarser grid. Time stepping also takes place on the coarse grid, providing further speed gains. Another important property of the method is that the basis functions are only computed once, and time savings are even greater when simulations are repeated for many source locations. We first present validation results for simple test models to demonstrate and quantify potential sources of error. These tests show that the fine-scale solution can be accurately approximated when the coarse grid applies a discretization up to four times larger than the original fine model. We then apply the multiscale algorithm to simulate a complete 2D seismic survey for a model with strong, fine-scale scatterers and apply standard migration algorithms to the resulting synthetic seismograms. The results again show small errors. Comparisons to a model that is upscaled by averaging densities on the fine grid show that the multiscale results are more accurate.

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Kumar, Pawan. "Aggregation based on graph matching and inexact coarse grid solve for algebraic two grid." International Journal of Computer Mathematics 91, no. 5 (August 15, 2013): 1061–81. http://dx.doi.org/10.1080/00207160.2013.821115.

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Mozgaleva, Marina L. "TWO-STAGE GRID METHOD OF SOLUTION OF BOUNDARY PROBLEMS OF STRUCTURAL MECHANICS WITH THE USE OF DISCRETE HAAR BASIS." International Journal for Computational Civil and Structural Engineering 13, no. 1 (March 22, 2017): 69–85. http://dx.doi.org/10.22337/2587-9618-2017-13-1-69-85.

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The distinctive paper is devoted to development of two-stage numerical method. At the first stage, the discrete problem is solved on a coarse grid, where the number of nodes in each direction is the same and is a pow-er of 2. Then the number of nodes in each direction is doubled and the resulting solution on a coarse grid using a discrete Haar basis is defined at the nodes of the fine grid as the initial approximation. At the second stage, we ob-tain a solution in the nodes of the fine grid using the most appropriate iterative method,. Test examples of the solu-tion of one-dimensional, two-dimensional and three-dimensional boundary problems are under consideration

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Lauritzen, Peter H., and Ramachandran D. Nair. "Monotone and Conservative Cascade Remapping between Spherical Grids (CaRS): Regular Latitude–Longitude and Cubed-Sphere Grids." Monthly Weather Review 136, no. 4 (April 1, 2008): 1416–32. http://dx.doi.org/10.1175/2007mwr2181.1.

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Abstract A high-order monotone and conservative cascade remapping algorithm between spherical grids (CaRS) is developed. This algorithm is specifically designed to remap between the cubed-sphere and regular latitude–longitude grids. The remapping approach is based on the conservative cascade method in which a two-dimensional remapping problem is split into two one-dimensional problems. This allows for easy implementation of high-order subgrid-cell reconstructions as well as the application of advanced monotone filters. The accuracy of CaRS is assessed by remapping analytic fields from the regular latitude–longitude grid to the cubed-sphere grid. In terms of standard error measures, CaRS is found to be competitive relative to an existing algorithm when regridding from a fine to a coarse grid and more accurate when regridding from a coarse to a fine grid.

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Shang, Yueqiang, and Jin Qin. "A Simplified Parallel Two-Level Iterative Method for Simulation of Incompressible Navier-Stokes Equations." Advances in Applied Mathematics and Mechanics 7, no. 6 (September 9, 2015): 715–35. http://dx.doi.org/10.4208/aamm.2014.m464.

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AbstractBased on two-grid discretization, a simplified parallel iterative finite element method for the simulation of incompressible Navier-Stokes equations is developed and analyzed. The method is based on a fixed point iteration for the equations on a coarse grid, where a Stokes problem is solved at each iteration. Then, on overlapped local fine grids, corrections are calculated in parallel by solving an Oseen problem in which the fixed convection is given by the coarse grid solution. Error bounds of the approximate solution are derived. Numerical results on examples of known analytical solutions, lid-driven cavity flow and backward-facing step flow are also given to demonstrate the effectiveness of the method.

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Mittet, Rune. "On the internal interfaces in finite-difference schemes." GEOPHYSICS 82, no. 4 (July 1, 2017): T159—T182. http://dx.doi.org/10.1190/geo2016-0477.1.

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Implementing sharp internal interfaces in finite-difference schemes with high spatial accuracy is challenging. The propagation of fields in a locally homogeneous part of a model can be performed with spectral accuracy. The implementations of interfaces are generally considered accurate to, at best, second order. This situation can be improved by a proper band limitation of the simulation grid. Interfaces can be located anywhere on the grid; however, the fine detail information regarding the interface location must be imprinted correctly in the coarse simulation grid. This can be done by starting out with a representation of a sharp material jump in the wavenumber domain and limiting the highest wavenumber to the maximum wavenumber allowed for the simulation grid. The resulting wavenumber representation is then transformed to the space domain. An alternative procedure is to create a fine grid model that is low-pass filtered to remove wavenumbers above the maximum wavenumber allowed for the coarse simulation grid. The fine grid is thereafter sampled at the required coordinates for the coarse simulation grid. An accurate and flexible interface implementation is a requisite for reducing staircase diffractions in higher dimensional finite-difference simulations. Our strategy achieves this. The frequency content of the source must be constrained to a level in which the spatial sampling is at approximately four to five grid points per shortest wavelength. Simulation results indicate that the implementation of the interface is accurate to at least the sixth order for large contrasts. Our method can be used for all systems of partial differential equations that formally can be expressed as a material parameter times a dynamic field on one side of the equal sign and with spatial derivatives on the other side of the equal sign. For geophysical simulations, the most important cases will be the Maxwell equations and the acoustic and elastic wave equations.

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Liu, Shang, and Yanping Chen. "A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations." Advances in Applied Mathematics and Mechanics 9, no. 3 (January 17, 2017): 757–74. http://dx.doi.org/10.4208/aamm.2015.m1370.

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AbstractIn the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using a expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy. An numerical example is also given to illustrate the effectiveness of the algorithm.

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Chen, Yanping, and Hanzhang Hu. "Two-Grid Method for Miscible Displacement Problem by Mixed Finite Element Methods and Mixed Finite Element Method of Characteristics." Communications in Computational Physics 19, no. 5 (May 2016): 1503–28. http://dx.doi.org/10.4208/cicp.scpde14.46s.

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AbstractThe miscible displacement of one incompressible fluid by another in a porous medium is governed by a system of two equations. One is elliptic form equation for the pressure and the other is parabolic form equation for the concentration of one of the fluids. Since only the velocity and not the pressure appears explicitly in the concentration equation, we use a mixed finite element method for the approximation of the pressure equation and mixed finite element method with characteristics for the concentration equation. To linearize the mixed-method equations, we use a two-grid algorithm based on the Newton iteration method for this full discrete scheme problems. First, we solve the original nonlinear equations on the coarse grid, then, we solve the linearized problem on the fine grid used Newton iteration once. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h = H2 in this paper. Finally, numerical experiment indicates that two-grid algorithm is very effective.

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Andrews, Peter N. Loezos, and Sankaran Sundaresan. "Coarse-Grid Simulation of Gas-Particle Flows in Vertical Risers." Industrial & Engineering Chemistry Research 44, no. 16 (August 2005): 6022–37. http://dx.doi.org/10.1021/ie0492193.

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Journal articles on the topic 'Coarse-grid'

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