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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. [ABSTRACT FROM AUTHOR]

In this paper, we first present the THex algorithm that refines a tetrahedral mesh into a hexahedral mesh. Strategies for efficient implementation of the THex algorithm are discussed. Then we present the lowest order weak Galerkin (WG) ( Q 0 , Q 0 ; RT [0] ) finite element method for solving the Darcy equation on general hexahedral meshes. This simple solver uses constant pressure unknowns inside hexahedra and on faces but specifies the discrete weak gradients of these basis functions in local Raviart-Thomas RT [0] spaces. Implementation of this solver is straightforward. The solver is locally mass-conservative, and produces continuous normal fluxes, regardless of hexahedral mesh quality. When the mesh is asymptotically parallelopiped, this Darcy solver exhibits optimal order convergence in pressure, velocity, and flux, as demonstrated by numerical results. [ABSTRACT FROM AUTHOR]

DarcyLite is a Matlab toolbox developed for numerical simulations of flow and transport in porous media in two dimensions. This paper focuses on the finite element methods and the corresponding code modules for solving the Darcy equation. Specifically, four major types of finite element solvers are presented: the continuous Galerkin (CG), the discontinuous Galerkin (DG), the weak Galerkin (WG), and the mixed finite element methods (MFEM). We further discuss the main design ideas and implementation strategies in DarcyLite. Numerical examples are included to demonstrate the usage and performance of this toolbox. [ABSTRACT FROM AUTHOR]

This paper presents a family of weak Galerkin finite element methods (WGFEMs) for Darcy flow computation. The WGFEMs are new numerical methods that rely on the novel concept of discrete weak gradients. The WGFEMs solve for pressure unknowns both in element interiors and on the mesh skeleton. The numerical velocity is then obtained from the discrete weak gradient of the numerical pressure. The new methods are quite different than many existing numerical methods in that they are locally conservative by design, the resulting discrete linear systems are symmetric and positive-definite, and there is no need for tuning problem-dependent penalty factors. We test the WGFEMs on benchmark problems to demonstrate the strong potential of these new methods in handling strong anisotropy and heterogeneity in Darcy flow. [ABSTRACT FROM AUTHOR]

This paper presents novel finite element solvers for Stokes flow that are pressure-robust due to the use of a lifting operator. Specifically, weak Galerkin (WG) finite element schemes are developed for the Stokes problem on quadrilateral and hexahedral meshes. Local Arbogast-Correa or Arbogast-Tao spaces are utilized for construction of discrete weak gradients. The lifting operator lifts WG test functions into H (div) -subspaces and removes pressure dependence of velocity errors. The pressure robustness of these solvers is validated theoretically and illustrated numerically. Comparison with the non-robust classical Taylor-Hood (Q 2 , Q 1) solver is presented. [ABSTRACT FROM AUTHOR]

This paper presents a lowest-order weak Galerkin (WG) finite element method for solving the Stokes equations on convex polygonal meshes. Constant vectors are used separately in element interiors and on edges to approximate fluid velocity, whereas constant scalars are used on elements to approximate the pressure. For the constant vector basis functions, their discrete weak gradients are established in a matrix space that is based on the C W 0 space (Chen and Wang, 2017), whereas their discrete weak divergences are calculated as elementwise constants. To circumvent the saddle-point problem, a reduced scheme for velocity is established by using three types of basis functions for the discretely divergence-free subspace. A procedure for subsequent pressure recovery is also developed. Error analysis along with numerical experiments on benchmarks are presented to demonstrate accuracy and efficiency of the proposed new method. • A simple but efficient new finite element method for Stokes flow. • The method applies to triangular, quadrilateral, and polygonal meshes in a unified way. • First ever use of H(div)-subspaces on polygons for Stokes flow. • Discretely divergence-free scheme and subsequent pressure recovery. • Numerical examples for Stokes flow on polygonal meshes. [ABSTRACT FROM AUTHOR]

This paper presents a novel 2-field finite element solver for linear poroelasticity on convex quadrilateral meshes. The Darcy flow is discretized for fluid pressure by a lowest-order weak Galerkin (WG) finite element method, which establishes the discrete weak gradient and numerical velocity in the lowest-order Arbogast–Correa space. The linear elasticity is discretized for solid displacement by the enriched Lagrangian finite elements with a special treatment for the volumetric dilation. These two types of finite elements are coupled through the implicit Euler temporal discretization to solve poroelasticity problems. A rigorous error analysis is presented along with numerical tests to demonstrate the accuracy and locking-free property of this new solver. • A FE solver for poroelasticity that solves for the two primal variables. • The solver uses the least degrees of freedom, compared to other existing methods. • The solver is locking-free. • Well-organized and easy-to-understand rigorous analysis. [ABSTRACT FROM AUTHOR]

This paper develops novel finite element solvers for linear poroelasticity problems on quadrilateral meshes. These solvers are based on the primal formulations of linear elasticity and Darcy flow. Specifically, the fluid pressure and solid displacement are approximated by scalar- or vector-valued polynomials of degree k ≥ 0 separately in element interiors and on edges. The discrete weak gradients of these shape functions are established in the broken (vector- or matrix-version) Arbogast–Correa spaces for approximations of the classical gradients in the variational forms. These weak Galerkin spatial discretizations are combined with the implicit Euler or Crank–Nicolson temporal discretizations to develop locking-free numerical solvers that have optimal order (k + 1) convergence rates in pressure, velocity, displacement, stress, and dilation. Rigorous analysis is presented and illustrated by numerical experiments on popular test cases. [ABSTRACT FROM AUTHOR]