Journal articles: 'Meshes convergence'

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Wu, Jinbiao, and Hui Zheng. "Uniform convergence of multigrid methods for adaptive meshes." Applied Numerical Mathematics 113 (March 2017): 109–23. http://dx.doi.org/10.1016/j.apnum.2016.11.005.

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Beir ao da Veiga, L., A. Chernov, L. Mascotto, and A. Russo. "Basic principles of hp virtual elements on quasiuniform meshes." Mathematical Models and Methods in Applied Sciences 26, no. 08 (2016): 1567–98. http://dx.doi.org/10.1142/s021820251650038x.

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In the present paper we initiate the study of [Formula: see text] Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size [Formula: see text] and in the polynomial degree [Formula: see text] in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included.

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Sun, Mei-Ling, and Shan Jiang. "Multiscale Basis Functions for Singular Perturbation on Adaptively Graded Meshes." Advances in Applied Mathematics and Mechanics 6, no. 5 (2014): 604–14. http://dx.doi.org/10.4208/aamm.2013.m488.

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AbstractWe apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes, which can provide a good balance between the numerical accuracy and computational cost. The multiscale space is built through standard finite element basis functions enriched with multiscale basis functions. The multiscale basis functions have abilities to capture originally perturbed information in the local problem, as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes, where the layer-adapted meshes are generated by a given parameter. Through numerical experiments we demonstrate that the multiscale method can acquire second order convergence in theL2norm and first order convergence in the energy norm on graded meshes, which is independent of ɛ. In contrast with the conventional methods, our method is much more accurate and effective.

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Huang, Junbin, and Klaus-Jürgen Bathe. "On the convergence of overlapping elements and overlapping meshes." Computers & Structures 244 (February 2021): 106429. http://dx.doi.org/10.1016/j.compstruc.2020.106429.

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Song, Songhe, Min Wan, Shengxi Wang, Desheng Wang, and Zhengping Zou. "Robust and Quality Boundary Constrained Tetrahedral Mesh Generation." Communications in Computational Physics 14, no. 5 (2013): 1304–21. http://dx.doi.org/10.4208/cicp.030612.010313a.

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AbstractA novel method for boundary constrained tetrahedral mesh generation is proposed based on Advancing Front Technique (AFT) and conforming Delaunay triangulation. Given a triangulated surface mesh, AFT is firstly applied to mesh several layers of elements adjacent to the boundary. The rest of the domain is then meshed by the conforming Delaunay triangulation. The non-conformal interface between two parts of meshes are adjusted. Mesh refinement and mesh optimization are then preformed to obtain a more reasonable-sized mesh with better quality. Robustness and quality of the proposed method is shown. Convergence proof of each stage as well as the whole algorithm is provided. Various numerical examples are included as well as the quality of the meshes.

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Stynes, Martin. "A Jejune Heuristic Mesh Theorem." Computational Methods in Applied Mathematics 3, no. 3 (2003): 488–92. http://dx.doi.org/10.2478/cmam-2003-0031.

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AbstractA singularly perturbed two-point boundary-value problem is considered. Working in the discrete maximum norm, a necessary condition for the convergence (uniformly in the singular perturbation parameter) of general difference schemes on general meshes is proved. This encompasses both a 1976 result of Miller for uniform meshes and more recent results of the same author that deal with piecewise uniform Shishkin meshes.

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Mei, Yanjie, Sulei Wang, Zhijie Xu, Chuanjing Song, and Yao Cheng. "Convergence Analysis of the LDG Method for Singularly Perturbed Reaction-Diffusion Problems." Symmetry 13, no. 12 (2021): 2291. http://dx.doi.org/10.3390/sym13122291.

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We analyse the local discontinuous Galerkin (LDG) method for two-dimensional singularly perturbed reaction–diffusion problems. A class of layer-adapted meshes, including Shishkin- and Bakhvalov-type meshes, is discussed within a general framework. Local projections and their approximation properties on anisotropic meshes are used to derive error estimates for energy and “balanced” norms. Here, the energy norm is naturally derived from the bilinear form of LDG formulation and the “balanced” norm is artificially introduced to capture the boundary layer contribution. We establish a uniform convergence of order k for the LDG method using the balanced norm with the local weighted L2 projection as well as an optimal convergence of order k+1 for the energy norm using the local Gauss–Radau projections. The numerical method, the layer structure as well as the used adaptive meshes are all discussed in a symmetry way. Numerical experiments are presented.

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Zhang, Jian Ming, and Yong He. "The Convergence of the H-P Version of the Finite Element Method with Quasi-Uniform Meshes for Three Dimensional Poisson Problems with Edge Singularity." Applied Mechanics and Materials 644-650 (September 2014): 1551–55. http://dx.doi.org/10.4028/www.scientific.net/amm.644-650.1551.

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This paper is concerned with the convergence of the h-p version of the finite element method for three dimensional Poisson problems with edge singularity on quasi-uniform meshes. First, we present the theoretical results for the convergence of the h-p version of the finite element method with quasi-uniform meshes for elliptic problems on polyhedral domains on smooth functions in the framework of Jacobi-weighted Sobolev spaces. Second, we investigate and analyze numerical results for three dimensional Poission problems with edge singularity. Finally, we verified the theoretical predictions by the numerical computation.

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Cangiani, A., P. Chatzipantelidis, G. Diwan, and E. H. Georgoulis. "Virtual element method for quasilinear elliptic problems." IMA Journal of Numerical Analysis 40, no. 4 (2019): 2450–72. http://dx.doi.org/10.1093/imanum/drz035.

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Abstract A virtual element method for the quasilinear equation $-\textrm{div} ({\boldsymbol \kappa }(u)\operatorname{grad} u)=f$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the $H^1$- and $L^2$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.

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Bojović, Dejan, and Boško Jovanović. "Fractional Order Convergence Rate Estimates Of Finite Difference Method On Nonuniform Meshes." Computational Methods in Applied Mathematics 1, no. 3 (2001): 213–21. http://dx.doi.org/10.2478/cmam-2001-0015.

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AbstractIn this paper we show how the theory of interpolation of function spaces can be used to establish convergence rate estimates for finite difference schemes on nonuniform meshes. As a model problem we consider the first boundary value problem for the Poisson equation. Using the interpolation theory we construct a fractional-order convergence rate estimate which is consistent with the smoothness of data.

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Youngquist, Jeremy, and Jörg Peters. "Solving Biharmonic Equations with Tri-Cubic C1 Splines on Unstructured Hex Meshes." Axioms 11, no. 11 (2022): 633. http://dx.doi.org/10.3390/axioms11110633.

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Unstructured hex meshes are partitions of three spaces into boxes that can include irregular edges, where n≠4 boxes meet along an edge, and irregular points, where the box arrangement is not consistent with a tensor-product grid. A new class of tri-cubic C1 splines is evaluated as a tool for solving elliptic higher-order partial differential equations over unstructured hex meshes. Convergence rates for four levels of refinement are computed for an implementation of the isogeometric Galerkin approach applied to Poisson’s equation and the biharmonic equation. The ratios of error are contrasted and superior to an implementation of Catmull-Clark solids. For the trivariate Poisson problem on irregularly partitioned domains, the reduction by 24 in the L2 norm is consistent with the optimal convergence on a regular grid, whereas the convergence rate for Catmull-Clark solids is measured as O(h3). The tri-cubic splines in the isogeometric framework correctly solve the trivariate biharmonic equation, but the convergence rate in the irregular case is lower than O(h4). An optimal reduction of 24 is observed when the functions on the C1 geometry are relaxed to be C0.

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12

Linß, Torsten. "Finite Difference Schemes for Convection-diffusion Problems with a Concentrated Source and a Discontinuous Convection Field." Computational Methods in Applied Mathematics 2, no. 1 (2001): 41–49. http://dx.doi.org/10.2478/cmam-2002-0003.

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AbstractA singularly perturbed convection-diffusion problem with a concentrated source is considered. The problem is solved numerically using two upwind difference schemes on general meshes. We prove convergence, uniformly with respect to the perturbation parameter, in the discrete maximum norm on Shishkin and Bakhvalov meshes. Numerical experiments complement our theoretical results.

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Sanchez, Richard, and Simone Santandrea. "Convergence Analysis for the Method of Characteristics in Unstructured Meshes." Nuclear Science and Engineering 183, no. 2 (2016): 196–213. http://dx.doi.org/10.13182/nse15-78.

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Mohamed, Mamdouh S., Anil N. Hirani, and Ravi Samtaney. "Numerical convergence of discrete exterior calculus on arbitrary surface meshes." International Journal for Computational Methods in Engineering Science and Mechanics 19, no. 3 (2018): 194–206. http://dx.doi.org/10.1080/15502287.2018.1446196.

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15

Selwood, P. "Convergence rates and classification for one-dimensional finite-element meshes." IMA Journal of Numerical Analysis 16, no. 1 (1996): 65–74. http://dx.doi.org/10.1093/imanum/16.1.65.

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AlSalem, H. J., P. Petrov, G. Newman, E. Um, and J. Rector. "Efficient discontinuous finite difference meshes for 3-D Laplace–Fourier domain seismic wavefield modelling in acoustic media with embedded boundaries." Geophysical Journal International 219, no. 2 (2019): 1253–67. http://dx.doi.org/10.1093/gji/ggz361.

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SUMMARY Simulation of acoustic wave propagation in the Laplace–Fourier (LF) domain, with a spatially uniform mesh, can be computationally demanding especially in areas with large velocity contrasts. To improve efficiency and convergence, we use 3-D second- and fourth-order velocity-pressure finite difference (FD) discontinuous meshes (DM). Our DM algorithm can use any spatial discretization ratio between meshes. We evaluate direct and iterative parallel solvers for computational speed, memory requirements and convergence. Benchmarks in realistic 3-D models and topographies show more efficient and stable results for DM with direct solvers than uniform mesh results with iterative solvers.

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17

Diogo, Teresa, Sean McKee, and Tao Tang. "Collocation methods for second-kind Volterra integral equations with weakly singular kernels." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 124, no. 2 (1994): 199–210. http://dx.doi.org/10.1017/s0308210500028432.

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In this paper it is shown that the use of uniform meshes leads to optimal convergence rates provided that the analytic solutions of a particular class of Volterra integral equations (VIEs) are smooth. If the exact solutions are not smooth, however, suitable transformations can be made so that the new VIEs possess smooth solutions. Spline collocation methods with uniform meshes applied to these new VIEs are then shown to be able to yield optimal (global) convergence rates. The general theory is applied to a typical case, i.e. the integral kernels consisting of the singular term (t − s) −½.

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Fridrich, David, Richard Liska, Ivan Tarant, Pavel Váchal, and Burton Wendroff. "CELL-CENTERED LAGRANGIAN LAX-WENDROFF HLL HYBRID SCHEME ON UNSTRUCTURED MESHES." Acta Polytechnica 61, SI (2021): 68–76. http://dx.doi.org/10.14311/ap.2021.61.0068.

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We have recently introduced a new cell-centered Lax-Wendroff HLL hybrid scheme for Lagrangian hydrodynamics [Fridrich et al. J. Comp. Phys. 326 (2016) 878-892] with results presented only on logical rectangular quadrilateral meshes. In this study we present an improved version on unstructured meshes, including uniform triangular and hexagonal meshes and non-uniform triangular and polygonal meshes. The performance of the scheme is verified on Noh and Sedov problems and its second-order convergence is verified on a smooth expansion test.Finally the choice of the scalar parameter controlling the amount of added artificial dissipation is studied.

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Kozhemyachenko, A. A., and A. V. Favorskaya. "Grid Convergence Analysis of Grid-Characteristic Method on Chimera Meshes in Ultrasonic Nondestructive Testing of Railroad Rail." Журнал вычислительной математики и математической физики 63, no. 10 (2023): 1687–705. http://dx.doi.org/10.31857/s0044466923100071.

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A three-dimensional direct problem of ultrasonic nondestructive testing of a railroad rail treated as a linear elastic medium is solved by applying a grid-characteristic method on curved structured Chimera and Cartesian background meshes. The algorithm involves mutual interpolation between Chimera and Cartesian meshes that takes into account the features of the transition from curved to Cartesian meshes in three-dimensional space. An analytical algorithm for generating Chimera meshes is proposed. The convergence of the developed numerical algorithms under mesh refinement in space is analyzed. A comparative analysis of the full-wave fields of the velocity modulus representing the propagation of a perturbation from its source is presented.

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Yang, Yan, Xiu Ye, and Shangyou Zhang. "A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids." Electronic Research Archive 32, no. 5 (2024): 3413–32. http://dx.doi.org/10.3934/era.2024158.

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<abstract><p>A pressure-robust stabilizer-free weak Galerkin (WG) finite element method has been defined for the Stokes equations on triangular and tetrahedral meshes. We have obtained pressure-independent error estimates for the velocity without any velocity reconstruction. The optimal-order convergence for the velocity of the WG approximation has been proved for the $ L^2 $ norm and the $ H^1 $ norm. The optimal-order error convergence has been proved for the pressure in the $ L^2 $ norm. The theory has been validated by performing some numerical tests on triangular and tetrahedral meshes.</p></abstract>

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Chen, Yanli, and Yonghai Li. "Optimal Bicubic Finite Volume Methods on Quadrilateral Meshes." Advances in Applied Mathematics and Mechanics 7, no. 4 (2015): 454–71. http://dx.doi.org/10.4208/aamm.2013.m401.

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AbstractIn this paper, an optimal bicubic finite volume method is established and analyzed for elliptic equations on quadrilateral meshes. Base on the so-called elementwise stiffness matrix analysis technique, we proceed the stability analysis. It is proved that the new scheme has optimal convergence rate in H1 norm. Additionally, we apply this analysis technique to bilinear finite volume method. Finally, numerical examples are provided to confirm the theoretical analysis of bicubic finite volume method.

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Coatléven, Julien. "Some multiple flow direction algorithms for overland flow on general meshes." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 6 (2020): 1917–49. http://dx.doi.org/10.1051/m2an/2020025.

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After recalling the most classical multiple flow direction algorithms (MFD), we establish their equivalence with a well chosen discretization of Manning–Strickler models for water flow. From this analogy, we derive a new MFD algorithm that remains valid on general, possibly non conforming meshes. We also derive a convergence theory for MFD algorithms based on the Manning–Strickler models. Numerical experiments illustrate the good behavior of the method even on distorted meshes.

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Dubois, François, and Pierre Lallemand. "On Triangular Lattice Boltzmann Schemes for Scalar Problems." Communications in Computational Physics 13, no. 3 (2013): 649–70. http://dx.doi.org/10.4208/cicp.381011.270112s.

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AbstractWe propose to extend the d’Humieres version of the lattice Boltzmann scheme to triangular meshes. We use Bravais lattices or more general lattices with the property that the degree of each internal vertex is supposed to be constant. On such meshes, it is possible to define the lattice Boltzmann scheme as a discrete particle method, without need of finite volume formulation or Delaunay-Voronoi hypothesis for the lattice. We test this idea for the heat equation and perform an asymptotic analysis with the Taylor expansion method for two schemes named D2T4 and D2T7. The results show a convergence up to second order accuracy and set new questions concerning a possible super-convergence.

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Morgado, M. Luísa, Magda Rebelo, and Luís L. Ferrás. "Stable and Convergent Finite Difference Schemes on NonuniformTime Meshes for Distributed-Order Diffusion Equations." Mathematics 9, no. 16 (2021): 1975. http://dx.doi.org/10.3390/math9161975.

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In this work, stable and convergent numerical schemes on nonuniform time meshes are proposed, for the solution of distributed-order diffusion equations. The stability and convergence of the numerical methods are proven, and a set of numerical results illustrate that the use of particular nonuniform time meshes provides more accurate results than the use of a uniform mesh, in the case of nonsmooth solutions.

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Aghili, Joubine, Daniele A. Di Pietro, and Berardo Ruffini. "An hp-Hybrid High-Order Method for Variable Diffusion on General Meshes." Computational Methods in Applied Mathematics 17, no. 3 (2017): 359–76. http://dx.doi.org/10.1515/cmam-2017-0009.

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AbstractIn this work, we introduce and analyze anhp-hybrid high-order (hp-HHO) method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We provehp-convergence estimates for both the energy- andL^{2}-norms of the error, which are the first of this kind for Hybrid High-Order methods. These results hinge on a novelhp-approximation lemma valid for general polytopal elements in arbitrary space dimension. The estimates are additionally fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on the square root of the local anisotropy, improving previous results for HHO methods. The expected exponential convergence behavior is numerically demonstrated on a variety of meshes for both isotropic and strongly anisotropic diffusion problems.

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Hashemi, M. S., J. Malekinagad, and H. R. Marasi. "Series Solution of the System of Fuzzy Differential Equations." Advances in Fuzzy Systems 2012 (2012): 1–16. http://dx.doi.org/10.1155/2012/407647.

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The homotopy analysis method (HAM) is proposed to obtain a semianalytical solution of the system of fuzzy differential equations (SFDE). The HAM contains the auxiliary parameterħ, which provides us with a simple way to adjust and control the convergence region of solution series. Concept ofħ-meshes and contour plots firstly are introduced in this paper which are the generations of traditionalh-curves. Convergency of this method for the SFDE has been considered and some examples are given to illustrate the efficiency and power of HAM.

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Leakey, Shannon, Vassilis Glenis, and Caspar Hewett. "Artificial Compressibility with Riemann Solvers: Convergence of Limiters on Unstructured Meshes." OpenFOAM® Journal 2 (March 4, 2022): 31–47. http://dx.doi.org/10.51560/ofj.v2.49.

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Free-surface flows and other variable density incompressible flows have numerous important applications in engineering.One way such flows can be modelled is to extend established numerical methods for compressible flows to incompressible flows using the method of artificial compressibility. Artificial compressibility introduces a pseudo-time derivative for pressure and, in each real-time step, the solution advances in pseudo-time until convergence to an incompressible limit - a fundamentally different approach than SIMPLE, PISO, and PIMPLE, the standard methods used in OpenFOAM. Although the artificial compressibility method is widespread in the literature, its application to free-surface flows is not. In this paper, we apply the method to variable density flows on 3D unstructured meshes for the first time, implementing a Godunov-type scheme with MUSCL reconstruction and Riemann solvers, where the free surface gets captured automatically by the contact wave in the Riemann solver. The critical problem in this implementation lies in the slope limiters used in the MUSCL reconstruction step. It is well-known that slope limiters can inhibit convergence to steady state on unstructured meshes; the problem is exacerbated here as convergence in pseudo-time is required not just once, but at every real-time step. We compare the limited gradient schemes included in OpenFOAM with an improved limiter from the literature, testing the solver against dam-break and hydrostatic pressure benchmarks. This work opens OpenFOAM up to the method of artificial compressibility, breaking the mould of PIMPLE and harnessing high-resolution shock-capturing schemes that are easier to parallelise.

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Righero, M., I. M. Bulai, M. A. Francavilla, et al. "Hierarchical Bases Preconditioner to Enhance Convergence of CFIE With Multiscale Meshes." IEEE Antennas and Wireless Propagation Letters 15 (2016): 1901–4. http://dx.doi.org/10.1109/lawp.2016.2542878.

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Xu, X., W. Huang, R. D. Russell, and J. F. Williams. "Convergence of de Boor's algorithm for the generation of equidistributing meshes." IMA Journal of Numerical Analysis 31, no. 2 (2010): 580–96. http://dx.doi.org/10.1093/imanum/drp052.

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Süli, Endre. "Convergence of Finite Volume Schemes for Poisson’s Equation on Nonuniform Meshes." SIAM Journal on Numerical Analysis 28, no. 5 (1991): 1419–30. http://dx.doi.org/10.1137/0728073.

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Chan, Tony F., and Jun Zou. "A convergence theory of multilevel additive Schwarz methods on unstructured meshes." Numerical Algorithms 13, no. 2 (1996): 365–98. http://dx.doi.org/10.1007/bf02207701.

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Joo, Younghwan, Heesun Choi, Gil-Eon Jeong, and Yonggyun Yu. "Dynamic graph-based convergence acceleration for topology optimization in unstructured meshes." Engineering Applications of Artificial Intelligence 132 (June 2024): 107916. http://dx.doi.org/10.1016/j.engappai.2024.107916.

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Bakhvalov, P. "Convergence Rate of the Spectral Difference Method on Regular Triangular Meshes." Lobachevskii Journal of Mathematics 45, no. 10 (2024): 4888–98. https://doi.org/10.1134/s1995080224604582.

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Bergot, Morgane, and Marc Duruflé. "Approximation of H(div) with High-Order Optimal Finite Elements for Pyramids, Prisms and Hexahedra." Communications in Computational Physics 14, no. 5 (2013): 1372–414. http://dx.doi.org/10.4208/cicp.120712.080313a.

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AbstractClassical facet elements do not provide an optimal rate of convergence of the numerical solution toward the solution of the exact problem in H(div-norm for general unstructured meshes containing hexahedra and prisms. We propose two new families of high-order elements for hexahedra, triangular prisms and pyramids that recover the optimal convergence. These elements have compatible restrictions with each other, such that they can be used directly on general hybrid meshes. Moreover the H(div) proposed spaces are completing the De Rham diagram with optimal elements previously constructed for H1 and H(curl) approximation. The obtained pyramidal elements are compared theoretically and numerically with other elements of the literature. Eventually, numerical results demonstrate the efficiency of the finite elements constructed.

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Brenner, Susanne C., Jintao Cui, and Li-yeng Sung. "Multigrid Methods Based on Hodge Decomposition for a Quad-Curl Problem." Computational Methods in Applied Mathematics 19, no. 2 (2019): 215–32. http://dx.doi.org/10.1515/cmam-2019-0011.

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AbstractIn this paper we investigate multigrid methods for a quad-curl problem on graded meshes. The approach is based on the Hodge decomposition. The solution for the quad-curl problem is approximated by solving standard second-order elliptic problems and optimal error estimates are obtained on graded meshes. We prove the uniform convergence of the multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results.

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SHI, Jiamin, Zhongshu LU, Luyi ZHANG, Sunjia LU, and Yao CHENG. "Uniform Convergence Analysis of the Discontinuous Galerkin Method on Layer-Adapted Meshes for Singularly Perturbed Problem." Wuhan University Journal of Natural Sciences 28, no. 5 (2023): 411–20. http://dx.doi.org/10.1051/wujns/2023285411.

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This paper concerns a discontinuous Galerkin (DG) method for a one-dimensional singularly perturbed problem which possesses essential characteristic of second order convection-diffusion problem after some simple transformations. We derive an optimal convergence of the DG method for eight layer-adapted meshes in a general framework. The convergence rate is valid independent of the small parameter. Furthermore, we establish a sharper L2-error estimate if the true solution has a special regular component. Numerical experiments are also given.

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Cho, Chien-Hong, and Chun-Yi Liu. "Convergence Analysis for a Three-Level Finite Difference Scheme of a Second Order Nonlinear ODE Blow-Up Problem." East Asian Journal on Applied Mathematics 7, no. 4 (2017): 679–96. http://dx.doi.org/10.4208/eajam.220816.300517a.

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AbstractWe consider the second order nonlinear ordinary differential equation u″ (t) = u1+α (α > 0) with positive initial data u(0) = a0, u′(0) = a1, whose solution becomes unbounded in a finite time T. The finite time T is called the blow-up time. Since finite difference schemes with uniform meshes can not reproduce such a phenomenon well, adaptively-defined grids are applied. Convergence with mesh sizes of certain smallness has been considered before. However, more iterations are required to obtain an approximate blow-up time if smaller meshes are applied. As a consequence, we consider in this paper a finite difference scheme with a rather larger grid size and show the convergence of the numerical solution and the numerical blow-up time. Application to the nonlinear wave equation is also discussed.

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Wada, Yoshitaka, Takuji Hayashi, Masanori Kikuchi, and Fei Xu. "Improvement of Unstructured Quadrilateral Mesh Quality for Multigrid Analysis." Advanced Materials Research 33-37 (March 2008): 833–38. http://dx.doi.org/10.4028/www.scientific.net/amr.33-37.833.

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Due to more complex and severe design restrictions, more effective and faster finite element analyses are demanded. There are several ways to compute FE analysis efficiently: parallel computing, fast iterative or direct solvers, adaptive analysis and so on. One of the most effective analysis ways is the combination of adaptive analysis and multigrid iterative solver, because an adaptive analysis requires several meshes with difference resolutions and multigrid solver utilizes such meshes to accelerate its computation. However, convergence of multigrid solver is largely affected by initial shape of each element. An effective mesh improvement method is proposed here. It is the combination of mesh coarsening and refinement. A good mesh can be obtained by the method to be applied to an initial mesh, and better convergence is achieved by the improved initial mesh.

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D’Angella, Davide, Stefan Kollmannsberger, Alessandro Reali, Ernst Rank, and Thomas J. R. Hughes. "An accurate strategy for computing reaction forces and fluxes on trimmed locally refined meshes." Journal of Mechanics 38 (2022): 60–76. http://dx.doi.org/10.1093/jom/ufac006.

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Abstract The finite element method is classically based on nodal Lagrange basis functions defined on conforming meshes. In this context, total reaction forces are commonly computed from the so-called “nodal forces”, yielding higher accuracy and convergence rates than reactions obtained from the differentiated primal solution (“direct” method). The finite cell method and isogeometric analysis promise to improve the interoperability of computer-aided design and computer-aided engineering, enabling a direct approach to the numerical simulation of trimmed geometries. However, body-unfitted meshes preclude the use of classic nodal reaction algorithms. This work shows that the direct method can perform particularly poorly for immersed methods. Instead, conservative reactions can be obtained from equilibrium expressions given by the weak problem formulation, yielding superior accuracy and convergence rates typical of nodal reactions. This approach is also extended to non-interpolatory basis functions, such as the (truncated) hierarchical B-splines.

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BOFFI, DANIELE, FUMIO KIKUCHI, and JOACHIM SCHÖBERL. "EDGE ELEMENT COMPUTATION OF MAXWELL'S EIGENVALUES ON GENERAL QUADRILATERAL MESHES." Mathematical Models and Methods in Applied Sciences 16, no. 02 (2006): 265–73. http://dx.doi.org/10.1142/s0218202506001145.

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Recent results prove that Nédélec edge elements do not achieve optimal rate of approximation on general quadrilateral meshes. In particular, lowest order edge elements provide stable but non-convergent approximation of Maxwell's eigenvalues. In this paper we analyze a modification of standard edge element that restores the optimality of the convergence. This modification is based on a projection technique that can be interpreted as a reduced integration procedure.

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Vieira, Gabriel Bancillon do Nascimento, Alice Rocha Pereira, and Sergio Koide. "Analysis of the Influence of the Numerical Mesh in the Hydrodynamic Modelling of Lake Paranoá Using Mike 3." Revista de Gestão Social e Ambiental 17, no. 10 (2023): e04050. http://dx.doi.org/10.24857/rgsa.v17n10-008.

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Purpose: This work aims to evaluate the effects of the different meshes constructed in MIKE 3 software on the simulation and calibration results of the model. Theoretical framework: 3D hydrodynamic models, such as MIKE 3, provide the closest representation of reality by simulating the gradients in the three spatial dimensions and solutioning the Navier-Stokes equations. In these models, meshes are used to represent complex geometries. An efficient computational mesh is required to allow convergence and stability of the solution of the equations and, furthermore, of the modelling result. Method/design/approach: Simulation of four meshes with distinct discretization, calibration, comparison, and assessment of the model performance for these four conceptual models considering: mesh’s number of elements, simulation time, mean absolute error (MAE), coefficient of determination (R2), and relative difference. Results and conclusions: For the meshes adopted for comparison, refinement only in the “throat” (region near the dam) did not show significant influences on the results that would justify its use, considering the high computational cost. Therefore, in this case, a sparse mesh and without refinement can be used in detriment of a mesh with refinement only in the “throat”. Research implication: Understand how different meshes discretization can significantly alter simulation time and highlight that optimized simulation requires an equilibrium between simulation time and mesh discretization to maintain model’s performance. Originality/value: Understanding and quantifying the influence of the discretization of the model's mesh on the simulation time and the performance of the model allows the optimization of the modeling, considering the cost-effectiveness of different discretizations leading to smaller simulation time with similar performance.

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Toprakseven, Suayip, and Seza Dinibutun. "A weak Galerkin finite element method for parabolic singularly perturbed convection-diffusion equations on layer-adapted meshes." Electronic Research Archive 32, no. 8 (2024): 5033–66. http://dx.doi.org/10.3934/era.2024232.

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<p>In this paper, we designed and analyzed a weak Galerkin finite element method on layer adapted meshes for solving the time-dependent convection-dominated problems. Error estimates for semi-discrete and fully-discrete schemes were presented, and the optimal order of uniform convergence has been obtained. A special interpolation was delicately designed based on the structures of the designed method and layer-adapted meshes. We provided various numerical examples to confirm the theoretical findings.</p>

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Khoei, Amir R., R. Yasbolaghi, and S. O. R. Biabanaki. "A polygonal-FEM technique in modeling large sliding contact on non-conformal meshes." Engineering Computations 32, no. 5 (2015): 1391–431. http://dx.doi.org/10.1108/ec-04-2014-0070.

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Purpose – In this paper, the polygonal-FEM technique is presented in modeling large deformation – large sliding contact on non-conformal meshes. The purpose of this paper is to present a new technique in modeling arbitrary interfaces and discontinuities for non-linear contact problems by capturing discontinuous deformations in elements cut by the contact surface in uniform non-conformal meshes. Design/methodology/approach – The geometry of contact surface is used to produce various polygonal elements at the intersection of the interface with the regular FE mesh, in which the extra degrees-of-freedom are defined along the interface. The contact constraints are imposed between polygonal elements produced along the contact surface through the node-to-surface contact algorithm. Findings – Numerical convergence analysis is carried out to study the convergence rate for various polygonal interpolation functions, including the Wachspress interpolation functions, the metric shape functions, the natural neighbor-based shape functions, and the mean value shape functions. Finally, numerical examples are solved to demonstrate the efficiency of proposed technique in modeling contact problems in large deformations. Originality/value – A new technique is presented based on the polygonal-FEM technique in modeling arbitrary interfaces and discontinuities for non-linear contact problems by capturing discontinuous deformations in elements cut by the contact surface in uniform non-conformal meshes.

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Liu, Dan, and Guoliang Xu. "Angle deficit approximation of Gaussian curvature and its convergence over quadrilateral meshes." Computer-Aided Design 39, no. 6 (2007): 506–17. http://dx.doi.org/10.1016/j.cad.2007.01.007.

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Brannick, James J., Hengguang Li, and Ludmil T. Zikatanov. "Uniform convergence of the multigridV-cycle on graded meshes for corner singularities." Numerical Linear Algebra with Applications 15, no. 2-3 (2008): 291–306. http://dx.doi.org/10.1002/nla.574.

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Alnashri, Yahya, and Hasan Alzubaidi. "Convergence of numerical schemes for convection–diffusion–reaction equations on generic meshes." Results in Applied Mathematics 19 (August 2023): 100379. http://dx.doi.org/10.1016/j.rinam.2023.100379.

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Wang, Xu, and Weiyin Ma. "Rational reparameterization of unstructured quadrilateral meshes for isogeometric analysis with optimal convergence." Computers & Mathematics with Applications 151 (December 2023): 304–25. http://dx.doi.org/10.1016/j.camwa.2023.09.050.

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Feischl, M., and Ch Schwab. "Exponential convergence in $$H^1$$ of hp-FEM for Gevrey regularity with isotropic singularities." Numerische Mathematik 144, no. 2 (2019): 323–46. http://dx.doi.org/10.1007/s00211-019-01085-z.

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AbstractFor functions $$u\in H^1(\Omega )$$u∈H1(Ω) in a bounded polytope $$\Omega \subset {\mathbb {R}}^d$$Ω⊂Rd$$d=1,2,3$$d=1,2,3 with plane sides for $$d=2,3$$d=2,3 which are Gevrey regular in $$\overline{\Omega }\backslash {\mathscr {S}}$$Ω¯\S with point singularities concentrated at a set $${\mathscr {S}}\subset \overline{\Omega }$$S⊂Ω¯ consisting of a finite number of points in $$\overline{\Omega }$$Ω¯, we prove exponential rates of convergence of hp-version continuous Galerkin finite element methods on affine families of regular, simplicial meshes in $$\Omega $$Ω. The simplicial meshes are geometrically refined towards $${\mathscr {S}}$$S but are otherwise unstructured.

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Apel, Thomas, Sergejs Rogovs, Johannes Pfefferer, and Max Winkler. "Maximum norm error estimates for Neumann boundary value problems on graded meshes." IMA Journal of Numerical Analysis 40, no. 1 (2018): 474–97. http://dx.doi.org/10.1093/imanum/dry076.

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AbstractThis paper deals with a priori pointwise error estimates for the finite element solution of boundary value problems with Neumann boundary conditions in polygonal domains. Due to the corners of the domain, the convergence rate of the numerical solutions can be lower than in the case of smooth domains. As a remedy, the use of local mesh refinement near the corners is considered. In order to prove quasi-optimal a priori error estimates, regularity results in weighted Sobolev spaces are exploited. This is the first work on the Neumann boundary value problem where both the regularity of the data is exactly specified and the sharp convergence order $h^{2} \lvert \ln h \rvert $ in the case of piecewise linear finite element approximations is obtained. As an extension we show the same rate for the approximate solution of a semilinear boundary value problem. The proof relies in this case on the supercloseness between the Ritz projection to the continuous solution and the finite element solution.

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Nhan, Thái Anh, and Relja Vulanović. "Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes." Advances in Numerical Analysis 2016 (February 28, 2016): 1–11. http://dx.doi.org/10.1155/2016/2161279.

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A one-dimensional linear convection-diffusion problem with a perturbation parameter ɛ multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is ɛ-uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “ɛ-uniform stability plus ɛ-uniform consistency implies ɛ-uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in ɛ. At the same time, the condition number of the discrete system becomes independent of ɛ due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when ɛ tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes.

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Journal articles on the topic 'Meshes convergence'

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