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Journal articles on the topic 'Raviart-Thomas space'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Bartels, Sören, and Zhangxian Wang. "Orthogonality relations of Crouzeix–Raviart and Raviart–Thomas finite element spaces." Numerische Mathematik 148, no. 1 (2021): 127–39. http://dx.doi.org/10.1007/s00211-021-01199-3.

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AbstractIdentities that relate projections of Raviart–Thomas finite element vector fields to discrete gradients of Crouzeix–Raviart finite element functions are derived under general conditions. Various implications such as discrete convex duality results and a characterization of the image of the projection of the Crouzeix–Ravaiart space onto elementwise constant functions are deduced.

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2

Kraus, J. K., and S. K. Tomar. "Algebraic multilevel iteration method for lowest order Raviart-Thomas space and applications." International Journal for Numerical Methods in Engineering 86, no. 10 (2011): 1175–96. http://dx.doi.org/10.1002/nme.3103.

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3

Zhu, Ailing. "Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems." Journal of Applied Mathematics 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/649468.

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The semidiscrete and fully discrete discontinuous mixed covolume schemes for the linear parabolic integrodifferential problems on triangular meshes are proposed. The error analysis of the semidiscrete and fully discrete discontinuous mixed covolume scheme is presented and the optimal order error estimate in discontinuousH(div)and first-order error estimate inL2are obtained with the lowest order Raviart-Thomas mixed element space.

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4

Swager, M. R., and Y. C. Zhou. "Genetic Exponentially Fitted Method for Solving Multi-dimensional Drift-diffusion Equations." Computational and Mathematical Biophysics 1 (March 20, 2013): 26–41. http://dx.doi.org/10.2478/mlbmb-2013-0001.

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AbstractA general approach was proposed in this article to develop high-order exponentially fitted basis functions for finite element approximations of multi-dimensional drift-diffusion equations for modeling biomolecular electrodiffusion processes. Such methods are highly desirable for achieving numerical stability and efficiency. We found that by utilizing the one-to-one correspondence between the continuous piecewise polynomial space of degree k + 1 and the divergencefree vector space of degree k, one can construct high-order two-dimensional exponentially fitted basis functions that are str

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5

Glowinski, Roland, and Serguei Lapin. "Solution of a Wave Equation by a Mixed Finite Element - Fictitious Domain Method." Computational Methods in Applied Mathematics 4, no. 4 (2004): 431–44. http://dx.doi.org/10.2478/cmam-2004-0024.

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AbstractThe main goal of this article is to investigate the capability of fictitious domain methods to simulate the scattering of linear waves by an obstacle whose shape does not fit the mesh. The space-time discretization relies on a combination of a mixed finite element method µa la Raviart-Thomas with a fairly standard finite difference scheme for the time discretization. The numerical results described in the article point to a good performance of the numerical method investigated here.

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6

Huang, Peiqi, Jinru Chen, and Mingchao Cai. "A Mortar Method Using Nonconforming and Mixed Finite Elements for the Coupled Stokes-Darcy Model." Advances in Applied Mathematics and Mechanics 9, no. 3 (2017): 596–620. http://dx.doi.org/10.4208/aamm.2016.m1397.

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AbstractIn this work, we study numerical methods for a coupled fluid-porous media flow model. The model consists of Stokes equations and Darcy's equations in two neighboring subdomains, coupling together through certain interface conditions. The weak form for the coupled model is of saddle point type. A mortar finite element method is proposed to approximate the weak form of the coupled problem. In our method, nonconforming Crouzeix-Raviart elements are applied in the fluid subdomain and the lowest order Raviart-Thomas elements are applied in the porous media subdomain; Meshes in different sub

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7

Bertrand, Fleurianne, Daniele Boffi, and Rolf Stenberg. "Asymptotically Exact A Posteriori Error Analysis for the Mixed Laplace Eigenvalue Problem." Computational Methods in Applied Mathematics 20, no. 2 (2020): 215–25. http://dx.doi.org/10.1515/cmam-2019-0099.

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AbstractThis paper derives a posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem. We discuss a reconstruction in the standard {H_{0}^{1}}-conforming space for the primal variable of the mixed Laplace eigenvalue problem and compare it with analogous approaches present in the literature for the corresponding source problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree, the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. Our reconstruction is performed lo

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8

BARRETT, JOHN W., and LEONID PRIGOZHIN. "A QUASI-VARIATIONAL INEQUALITY PROBLEM IN SUPERCONDUCTIVITY." Mathematical Models and Methods in Applied Sciences 20, no. 05 (2010): 679–706. http://dx.doi.org/10.1142/s0218202510004404.

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We derive a class of analytical solutions and a dual formulation of a scalar two-space-dimensional quasi-variational inequality problem in applied superconductivity. We approximate this formulation by a fully practical finite element method based on the lowest order Raviart–Thomas element, which yields approximations to both the primal and dual variables (the magnetic and electric fields). We prove the subsequence convergence of this approximation, and hence prove the existence of a solution to both the dual and primal formulations, for strictly star-shaped domains. The effectiveness of the ap

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9

Diogene Vianney, Pongui ngoma, Nguimbi Germain, and Likibi Pellat Rhoss Beaunheur. "The effect of numerical integration in mixed finite element approximation in the simulation of miscible displacement." International Journal of Applied Mathematical Research 6, no. 2 (2017): 44. http://dx.doi.org/10.14419/ijamr.v6i2.7320.

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We consider the effect of numerical integration in finite element procedures applied to a nonlinear system of two coupled partial differential equations describing the miscible displacement of one incompressible fluid by another in a porous meduim. We consider the use of the numerical quadrature scheme for approximating the pressure and velocity by a mixed method using Raviart - Thomas space of index and the concentration by a standard Galerkin method. We also give some sufficient conditions on the quadrature scheme to ensure that the order of convergence is unaltered in the presence of numeri

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10

Gillette, Andrew, Alexander Rand, and Chandrajit Bajaj. "Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes." Computational Methods in Applied Mathematics 16, no. 4 (2016): 667–83. http://dx.doi.org/10.1515/cmam-2016-0019.

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AbstractWe combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart–Thomas, and Brezzi–Douglas–Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space wi

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11

Arnold, Douglas N., and Richard S. Falk. "Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model." Mathematical Models and Methods in Applied Sciences 07, no. 02 (1997): 217–38. http://dx.doi.org/10.1142/s0218202597000141.

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An analysis is presented for a recently proposed finite element method for the Reissner–Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual "locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t = O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numeri

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12

Bertrand, Fleurianne, Marcel Moldenhauer, and Gerhard Starke. "A Posteriori Error Estimation for Planar Linear Elasticity by Stress Reconstruction." Computational Methods in Applied Mathematics 19, no. 3 (2019): 663–79. http://dx.doi.org/10.1515/cmam-2018-0004.

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AbstractThe nonconforming triangular piecewise quadratic finite element space by Fortin and Soulie can be used for the displacement approximation and its combination with discontinuous piecewise linear pressure elements is known to constitute a stable combination for incompressible linear elasticity computations. In this contribution, we extend the stress reconstruction procedure and resulting guaranteed a posteriori error estimator developed by Ainsworth, Allendes, Barrenechea and Rankin [2] and by Kim [18] to linear elasticity. In order to get a guaranteed reliability bound with respect to t

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13

Araya, Rodolfo, Rodolfo Rodríguez, and Pablo Venegas. "Numerical analysis of a time domain elastoacoustic problem." IMA Journal of Numerical Analysis 40, no. 2 (2019): 1122–53. http://dx.doi.org/10.1093/imanum/dry093.

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Abstract This paper deals with the numerical analysis of a system of second order in time partial differential equations modeling the vibrations of a coupled system that consists of an elastic solid in contact with an inviscid compressible fluid. We analyze a weak formulation with the unknowns in both media being the respective displacement fields. For its numerical approximation, we propose first a semidiscrete in space discretization based on standard Lagrangian elements in the solid and Raviart–Thomas elements in the fluid. We establish its well-posedness and derive error estimates in appro

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14

Huang, Xuehai. "A Reduced Local Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity." Mathematical Problems in Engineering 2013 (2013): 1–11. http://dx.doi.org/10.1155/2013/546408.

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A reduced local discontinuous Galerkin (RLDG) method for nearly incompressible linear elasticity is proposed in this paper, which is locking-free. RLDG method can be formally regarded as a special case of LDG method withC11=0. However, RLDG method is actually not covered by LDG method, whereC11must be chosen to be positive to ensure the stability of LDG method. RLDG method can also be considered as the localization of some symmetric nonconforming mixed finite element method. The implementation of RLDG method is discussed. By introducing a lifting operator as LDG method, RLDG method can be rewr

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15

Selzer, Philipp, and Olaf A. Cirpka. "Postprocessing of standard finite element velocity fields for accurate particle tracking applied to groundwater flow." Computational Geosciences 24, no. 4 (2020): 1605–24. http://dx.doi.org/10.1007/s10596-020-09969-y.

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Abstract Particle tracking is a computationally advantageous and fast scheme to determine travel times and trajectories in subsurface hydrology. Accurate particle tracking requires element-wise mass-conservative, conforming velocity fields. This condition is not fulfilled by the standard linear Galerkin finite element method (FEM). We present a projection, which maps a non-conforming, element-wise given velocity field, computed on triangles and tetrahedra, onto a conforming velocity field in lowest-order Raviart-Thomas-Nédélec ($\mathcal {RTN}_{0}$ R T N 0 ) space, which meets the requirements

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16

Hu, Jun, та Shangyou Zhang. "Finite element approximations of symmetric tensors on simplicial grids in ℝn: The lower order case". Mathematical Models and Methods in Applied Sciences 26, № 09 (2016): 1649–69. http://dx.doi.org/10.1142/s0218202516500408.

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In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order [Formula: see text] from [Finite element approximations of symmetric tensors on simplicial grids in [Formula: see text]: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each [Formula: see text]-dimensional simplex, by [Formula: see text] face bubble functions in the symmetric tensor finite element space of

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17

Jamei, Mehdi, and H. Ghafouri. "An efficient discontinuous Galerkin method for two-phase flow modeling by conservative velocity projection." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 1 (2016): 63–84. http://dx.doi.org/10.1108/hff-08-2014-0247.

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Purpose – The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection. Design/methodology/approach – The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weight

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18

Jamei, Mehdi, and H. Ghafouri. "A novel discontinuous Galerkin model for two-phase flow in porous media using an improved IMPES method." International Journal of Numerical Methods for Heat & Fluid Flow 26, no. 1 (2016): 284–306. http://dx.doi.org/10.1108/hff-01-2015-0008.

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Purpose – The purpose of this paper is to present an efficient improved version of Implicit Pressure-Explicit Saturation (IMPES) method for the solution of incompressible two-phase flow model based on the discontinuous Galerkin (DG) numerical scheme. Design/methodology/approach – The governing equations, based on the wetting-phase pressure-saturation formulation, are discretized using various primal DG schemes. The authors use H(div) velocity reconstruction in Raviart-Thomas space (RT_0 and RT_1), the weighted average formulation, and the scaled penalties to improve the spatial discretization.

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19

Lu, Zuliang, Yanping Chen, and Weishan Zheng. "A Posteriori Error Estimates of Lowest Order Raviart-Thomas Mixed Finite Element Methods for Bilinear Optimal Control Problems." East Asian Journal on Applied Mathematics 2, no. 2 (2012): 108–25. http://dx.doi.org/10.4208/eajam.130212.300312a.

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AbstractA Raviart-Thomas mixed finite element discretization for general bilinear optimal control problems is discussed. The state and co-state are approximated by lowest order Raviart-Thomas mixed finite element spaces, and the control is discretized by piecewise constant functions. A posteriori error estimates are derived for both the coupled state and the control solutions, and the error estimators can be used to construct more efficient adaptive finite element approximations for bilinear optimal control problems. An adaptive algorithm to guide the mesh refinement is also provided. Finally,

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20

Caucao, Sergio, Gabriel N. Gatica, and Ricardo Oyarzúa. "Analysis of an augmented fully-mixed formulation for the coupling of the Stokes and heat equations." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 5 (2018): 1947–80. http://dx.doi.org/10.1051/m2an/2018027.

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We introduce and analyse an augmented mixed variational formulation for the coupling of the Stokes and heat equations. More precisely, the underlying model consists of the Stokes equation suggested by the Oldroyd model for viscoelastic flow, coupled with the heat equation through a temperature-dependent viscosity of the fluid and a convective term. The original unknowns are the polymeric part of the extra-stress tensor, the velocity, the pressure, and the temperature of the fluid. In turn, for convenience of the analysis, the strain tensor, the vorticity, and an auxiliary symmetric tensor are

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21

Lashuk, I. V., and P. S. Vassilevski. "Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties." Numerical Linear Algebra with Applications 19, no. 2 (2012): 414–26. http://dx.doi.org/10.1002/nla.1819.

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22

Hou, Tianliang, and Li Li. "Error Estimates of Mixed Methods for Optimal Control Problems Governed by General Elliptic Equations." Advances in Applied Mathematics and Mechanics 8, no. 6 (2016): 1050–71. http://dx.doi.org/10.4208/aamm.2014.m807.

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AbstractIn this paper, we investigate the error estimates of mixed finite element methods for optimal control problems governed by general elliptic equations. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We derive L2 and H–1-error estimates both for the control variable and the state variables. Finally, a numerical example is given to demonstrate the theoretical results.

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23

Manickam, K., and P. Prakash. "Mixed Finite Element Methods for Fourth Order Elliptic Optimal Control Problems." Numerical Mathematics: Theory, Methods and Applications 9, no. 4 (2016): 528–48. http://dx.doi.org/10.4208/nmtma.2016.m1405.

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AbstractIn this paper, a priori error estimates are derived for the mixed finite element discretization of optimal control problems governed by fourth order elliptic partial differential equations. The state and co-state are discretized by Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. The error estimates derived for the state variable as well as those for the control variable seem to be new. We illustrate with a numerical example to confirm our theoretical results.

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24

Chen, Yanping, Tianliang Hou, and Weishan Zheng. "Error Estimates and Superconvergence of Mixed Finite Element Methods for Optimal Control Problems with Low Regularity." Advances in Applied Mathematics and Mechanics 4, no. 06 (2012): 751–68. http://dx.doi.org/10.4208/aamm.12-12s05.

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AbstractIn this paper, we investigate the error estimates and superconvergence property of mixed finite element methods for elliptic optimal control problems. The state and co-state are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. We deriveL2andL∞-error estimates for the control variable. Moreover, using a recovery operator, we also derive some superconvergence results for the control variable. Finally, a numerical example is given to demonstrate the theoretical results.

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25

Chen, Yanping, and Zhuoqing Lin. "A Posteriori Error Estimates of Semidiscrete Mixed Finite Element Methods for Parabolic Optimal Control Problems." East Asian Journal on Applied Mathematics 5, no. 1 (2015): 85–108. http://dx.doi.org/10.4208/eajam.010314.110115a.

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AbstractA posteriori error estimates of semidiscrete mixed finite element methods for quadratic optimal control problems involving linear parabolic equations are developed. The state and co-state are discretised by Raviart-Thomas mixed finite element spaces of order k, and the control is approximated by piecewise polynomials of order k (k ≥ 0). We derive our a posteriori error estimates for the state and the control approximations via a mixed elliptic reconstruction method. These estimates seem to be unavailable elsewhere in the literature, although they represent an important step towards dev

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26

Lu, Zuliang. "Adaptive Mixed Finite Element Methods for Parabolic Optimal Control Problems." Mathematical Problems in Engineering 2011 (2011): 1–21. http://dx.doi.org/10.1155/2011/217493.

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We will investigate the adaptive mixed finite element methods for parabolic optimal control problems. The state and the costate are approximated by the lowest-order Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant elements. We derive a posteriori error estimates of the mixed finite element solutions for optimal control problems. Such a posteriori error estimates can be used to construct more efficient and reliable adaptive mixed finite element method for the optimal control problems. Next we introduce an adaptive algorithm to guide the mesh refi

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27

Ambartsumyan, Ilona, Eldar Khattatov, Jeonghun J. Lee, and Ivan Yotov. "Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra." Mathematical Models and Methods in Applied Sciences 29, no. 06 (2019): 1037–77. http://dx.doi.org/10.1142/s0218202519500167.

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We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system fo

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28

Almonacid, Javier A., Hugo S. Díaz, Gabriel N. Gatica, and Antonio Márquez. "A fully mixed finite element method for the coupling of the Stokes and Darcy–Forchheimer problems." IMA Journal of Numerical Analysis 40, no. 2 (2019): 1454–502. http://dx.doi.org/10.1093/imanum/dry099.

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Abstract In this paper we introduce and analyze a fully mixed formulation for the nonlinear problem given by the coupling of the Stokes and Darcy–Forchheimer equations with the Beavers–Joseph–Saffman condition on the interface. This new approach yields non-Hilbert normed spaces and a twofold saddle point structure for the corresponding operator equation, whose continuous and discrete solvabilities are analyzed by means of a suitable abstract theory developed for this purpose. In particular, feasible choices of finite element subspaces include PEERS of the lowest order for the stress of the flu

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29

Caucao, Sergio, Gabriel N. Gatica, Ricardo Oyarzúa, and Felipe Sandoval. "Residual-based a posteriori error analysis for the coupling of the Navier–Stokes and Darcy–Forchheimer equations." ESAIM: Mathematical Modelling and Numerical Analysis 55, no. 2 (2021): 659–87. http://dx.doi.org/10.1051/m2an/2021005.

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In this paper we consider a mixed variational formulation that have been recently proposed for the coupling of the Navier–Stokes and Darcy–Forchheimer equations, and derive, though in a non-standard sense, a reliable and efficient residual-based a posteriori error estimator suitable for an adaptive mesh-refinement method. For the reliability estimate, which holds with respect to the square root of the error estimator, we make use of the inf-sup condition and the strict monotonicity of the operators involved, a suitable Helmholtz decomposition in non-standard Banach spaces in the porous medium,

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Bertrand, Fleurianne, Zhiqiang Cai, and Eun Young Park. "Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry." Computational Methods in Applied Mathematics 19, no. 3 (2019): 415–30. http://dx.doi.org/10.1515/cmam-2018-0255.

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AbstractThis paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the{L^{2}}norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares

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31

Almonacid, Javier A., and Gabriel N. Gatica. "A Fully-Mixed Finite Element Method for the n-Dimensional Boussinesq Problem with Temperature-Dependent Parameters." Computational Methods in Applied Mathematics 20, no. 2 (2020): 187–213. http://dx.doi.org/10.1515/cmam-2018-0187.

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AbstractIn this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of n-dimensional fluids, {n\in\{2,3\}}, with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in a

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Alvarez, Mario, Gabriel N. Gatica, and Ricardo Ruiz-Baier. "A mixed-primal finite element approximation of a sedimentation–consolidation system." Mathematical Models and Methods in Applied Sciences 26, no. 05 (2016): 867–900. http://dx.doi.org/10.1142/s0218202516500202.

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This paper is devoted to the mathematical and numerical analysis of a strongly coupled flow and transport system typically encountered in continuum-based models of sedimentation–consolidation processes. The model focuses on the steady-state regime of a solid–liquid suspension immersed in a viscous fluid within a permeable medium, and the governing equations consist in the Brinkman problem with variable viscosity, written in terms of Cauchy pseudo-stresses and bulk velocity of the mixture; coupled with a nonlinear advection — nonlinear diffusion equation describing the transport of the solids v

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Voronin, Kirill, and Yuri Laevsky. "A new approach to constructing vector splitting schemes in mixed finite element method for parabolic problems." Journal of Numerical Mathematics 25, no. 1 (2017). http://dx.doi.org/10.1515/jnma-2015-0076.

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AbstractA general setting for a new approach to constructing vector splitting schemes in mixed FEM for heat transfer problem is considered. For space approximation Raviart–Thomas finite elements of lowest order are implemented on rectangular (parallelepiped) mesh. The main question discussed is how to implement time discretization so as to obtain efficient numerical algorithms.The key idea of the proposed approach is to use scalar splitting schemes for heat flux divergence. This allows one to carry out accuracy and stability analysis on the basis of the well-known results for underlying scalar

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34

Caucao, Sergio, and Ivan Yotov. "A Banach space mixed formulation for the unsteady Brinkman–Forchheimer equations." IMA Journal of Numerical Analysis, August 19, 2020. http://dx.doi.org/10.1093/imanum/draa035.

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Abstract We propose and analyse a mixed formulation for the Brinkman–Forchheimer equations for unsteady flows. Our approach is based on the introduction of a pseudostress tensor related to the velocity gradient and pressure, leading to a mixed formulation where the pseudostress tensor and the velocity are the main unknowns of the system. We establish existence and uniqueness of a solution to the weak formulation in a Banach space setting, employing classical results on nonlinear monotone operators and a regularization technique. We then present well posedness and error analysis for semidiscret

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Muñoz, Guillermo, Edmundo Del Valle, and Armando M. Gómez-Torres. "Canonical Implementation of Simplified Spherical Harmonics (SPL) in the Neutron Diffusion Code AZNHEX." Journal of Nuclear Engineering and Radiation Science, December 8, 2020. http://dx.doi.org/10.1115/1.4049277.

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Abstract On the basis of the neutron transport equation, several neutron diffusion codes were developed in the past due to its simplified form and the limitations on computational power. Numerical schemes were developed to solve the diffusion equation with acceptable performance but still with problems in the regions where diffusion theory is not physically accurate. Nowadays, computational power has significantly increased and thus, new numerical schemes are being implemented to deal directly with neutron transport equation. The Simplified Spherical Harmonics approximation can be found among

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Le Meledo, Elise, Philipp Öffner, and Remi Abgrall. "General polytopal H(div)-conformal finite elements and their discretisation spaces." ESAIM: Mathematical Modelling and Numerical Analysis, July 22, 2020. http://dx.doi.org/10.1051/m2an/2020048.

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We present a class of discretisation spaces and H(div) - conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the Raviart - Thomas elements on the boundaries, the designed frameworks offer a wide range of H(div) - conformal discretisations. As those elements are set up through degrees of freedom, their definitions are easily amenable to the properties the approximated quantities are wished to fulfil. Furthermore, we show that one straightforward restriction of this general sett

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Eugenia Cejas, María, Ricardo Durán, and Mariana Prieto. "Mixed methods for degenerate elliptic problems and application to fractional Laplacian." ESAIM: Mathematical Modelling and Numerical Analysis, September 13, 2020. http://dx.doi.org/10.1051/m2an/2020068.

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We analyze the approximation by mixed finite element methods of solutions of equations of the form div [[EQUATION]] , where the coefficient a=a(x) can degenerate going to zero or infinity. First, we extend the classic error analysis to this case provided that the coefficient $a$ belongs to the Muckenhoupt class [[EQUATION]] . The analysis developed applies to general mixed finite element spaces satisfying the standard commutative diagram property, whenever some stability and interpolation error estimates are valid in weighted norms. Next, we consider in detail the case of Raviart-Thomas spaces

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Gao, Huadong, Weiwei Sun, and Chengda Wu. "Optimal error estimates and recovery technique of a mixed finite element method for nonlinear thermistor equations." IMA Journal of Numerical Analysis, September 2, 2020. http://dx.doi.org/10.1093/imanum/draa063.

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Abstract This paper is concerned with optimal error estimates and recovery technique of a classical mixed finite element method for the thermistor problem, which is governed by a parabolic/elliptic system with strong nonlinearity and coupling. The method is based on a popular combination of the lowest-order Raviart–Thomas mixed approximation for the electric potential/field $(\phi , \boldsymbol{\theta })$ and the linear Lagrange approximation for the temperature $u$. A common question is how the first-order approximation influences the accuracy of the second-order approximation to the temperat

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Gatica, Gabriel N., Salim Meddahi, and Ricardo Ruiz-Baier. "An Lp spaces-based formulation yielding a new fully mixed finite element method for the coupled Darcy and heat equations." IMA Journal of Numerical Analysis, September 10, 2021. http://dx.doi.org/10.1093/imanum/drab063.

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Abstract In this work we present and analyse a new fully mixed finite element method for the nonlinear problem given by the coupling of the Darcy and heat equations. Besides the velocity, pressure and temperature variables of the fluid, our approach is based on the introduction of the pseudoheat flux as a further unknown. As a consequence of it, and due to the convective term involving the velocity and the temperature, we arrive at saddle point-type schemes in Banach spaces for both equations. In particular, and as suggested by the solvability of a related Neumann problem to be employed in the

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