The lowest-order weak Galerkin finite element method for the Darcy equation on quadrilateral and hybrid meshes

Abstract This paper investigates the lowest-order weak Galerkin finite element method for solving the Darcy equation on quadrilateral and hybrid meshes consisting of quadrilaterals and triangles. In this approach, the pressure is approximated by constants in element interiors and on edges. The discrete weak gradients of these constant basis functions are specified in local Raviart–Thomas spaces, specifically R T 0 for triangles and unmapped R T [ 0 ] for quadrilaterals. These discrete weak gradients are used to approximate the classical gradient when solving the Darcy equation. The method produces continuous normal fluxes and is locally mass-conservative, regardless of mesh quality, and has optimal order convergence in pressure, velocity, and normal flux, when the quadrilaterals are asymptotically parallelograms. Implementation is straightforward and results in symmetric positive-definite discrete linear systems. We present numerical experiments and comparisons with other existing methods.

[1]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[2]  Mary F. Wheeler,et al.  A Multipoint Flux Mixed Finite Element Method , 2006, SIAM J. Numer. Anal..

[3]  Richard E. Ewing,et al.  The Mathematics of Reservoir Simulation , 2016 .

[4]  Haiying Wang,et al.  Locally Conservative Fluxes for the Continuous Galerkin Method , 2007, SIAM J. Numer. Anal..

[5]  Sean McGinty,et al.  A decade of modelling drug release from arterial stents. , 2014, Mathematical biosciences.

[6]  Todd Arbogast,et al.  Two Families of H(div) Mixed Finite Elements on Quadrilaterals of Minimal Dimension , 2016, SIAM J. Numer. Anal..

[7]  Guang Lin,et al.  Weak Galerkin finite element methods for Darcy flow: Anisotropy and heterogeneity , 2014, J. Comput. Phys..

[8]  Douglas N. Arnold,et al.  Quadrilateral H(div) Finite Elements , 2004, SIAM J. Numer. Anal..

[9]  Béatrice Rivière,et al.  Discontinuous Galerkin Methods for Second-Order Elliptic PDE with Low-Regularity Solutions , 2011, J. Sci. Comput..

[10]  Victor M. Calo,et al.  Mathematical modeling of coupled drug and drug-encapsulated nanoparticle transport in patient-specific coronary artery walls , 2012 .

[11]  T. F. Russell.  Relationships among Some Conservative Discretization Methods , 2000 .

[12]  So-Hsiang Chou,et al.  On the regularity and uniformness conditions on quadrilateral grids , 2002 .

[13]  Junping Wang,et al.  A weak Galerkin finite element method with polynomial reduction , 2013, J. Comput. Appl. Math..

[14]  Yaoxin Zhang,et al.  Hybrid mesh generation using advancing reduction technique , 2015 .

[15]  Lin Mu,et al.  Weak Galerkin Finite Element Methods for Second-Order Elliptic Problems on Polytopal Meshes , 2012 .

[16]  Zhuoran Wang,et al.  DarcyLite: A Matlab Toolbox for Darcy Flow Computation , 2016, ICCS.

[17]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[18]  J. Remacle,et al.  Blossom‐Quad: A non‐uniform quadrilateral mesh generator using a minimum‐cost perfect‐matching algorithm , 2012 .

[19]  C. Bahriawati,et al.  Three Matlab Implementations of the Lowest-order Raviart-Thomas Mfem with a Posteriori Error Control , 2005 .

[20]  Do Y. Kwak,et al.  Mixed finite element methods for general quadrilateral grids , 2011, Appl. Math. Comput..

[21]  Carsten Carstensen,et al.  Remarks around 50 lines of Matlab: short finite element implementation , 1999, Numerical Algorithms.

[22]  L. Durlofsky.  Accuracy of mixed and control volume finite element approximations to Darcy velocity and related quantities , 1994 .

[23]  B. Rivière,et al.  Superconvergence and H(div) projection for discontinuous Galerkin methods , 2003 .

[24]  Graham F. Carey,et al.  An enhanced polygonal finite‐volume method for unstructured hybrid meshes , 2007 .

[25]  Q. Zou,et al.  Unified analysis of higher-order finite volume methods for parabolic problems on quadrilateral meshes , 2016 .

[26]  Mary F. Wheeler,et al.  A multipoint flux mixed finite element method on distorted quadrilaterals and hexahedra , 2011, Numerische Mathematik.

[27]  Junping Wang,et al.  A weak Galerkin finite element method for second-order elliptic problems , 2011, J. Comput. Appl. Math..

[28]  Bin Zheng,et al.  The THex Algorithm and a Simple Darcy Solver on Hexahedral Meshes , 2017, ICCS.

[29]  Shuyu Sun,et al.  A Locally Conservative Finite Element Method Based on Piecewise Constant Enrichment of the Continuous Galerkin Method , 2009, SIAM J. Sci. Comput..

[30]  Douglas N. Arnold,et al.  Approximation by quadrilateral finite elements , 2000, Math. Comput..

[31]  Richard E. Ewing,et al.  Superconvergence of Mixed Finite Element Approximations over Quadrilaterals , 1993 .

[32]  Axel Modave,et al.  GPU-accelerated discontinuous Galerkin methods on hybrid meshes , 2015, J. Comput. Phys..

[33]  Guang Lin,et al.  A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods , 2015, J. Comput. Appl. Math..

[34]  Guang Lin,et al.  On Application of the Weak Galerkin Finite Element Method to a Two-Phase Model for Subsurface Flow , 2016, J. Sci. Comput..

[35]  Victor Ginting,et al.  On the Application of the Continuous Galerkin Finite Element Method for Conservation Problems , 2013, SIAM J. Sci. Comput..

[36]  Weizhang Huang,et al.  Discrete Maximum Principle for the Weak Galerkin Method for Anisotropic Diffusion Problems , 2014, 1401.6232.

[37]  YE JUNPINGWANGANDXIU.  A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR SECOND ORDER ELLIPTIC PROBLEMS , 2014 .

[38]  Lin Mu,et al.  Journal of Computational and Applied Mathematics Convergence of the Discontinuous Finite Volume Method for Elliptic Problems with Minimal Regularity , 2022 .