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.
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 (Q2,Q1) solver is presented.
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.
This paper presents a finite element method for solving coupled Stokes–Darcy flow problems by combining the classical Bernardi–Raugel finite elements and the recently developed Arbogast–Correa (AC) ...spaces on quadrilateral meshes. The novel weak Galerkin methodology is employed for discretization of the Darcy equation. Specifically, piecewise constant approximants separately defined in element interiors and on edges are utilized to approximate the Darcy pressure. The discrete weak gradients of these shape functions and the numerical Darcy velocity are established in the lowest order AC space. The Bernardi–Raugel elements (BR1,Q0) are used to discretize the Stokes equations. These two types of discretizations are combined at an interface, where kinematic, normal stress, and the Beavers–Joseph–Saffman (BJS) conditions are applied. Rigorous error analysis along with numerical experiments demonstrate that the method is stable and has optimal-order accuracy.
•A new finite element method with the least unknowns.•Weak Galerkin pressure unknowns on interface edges behaving similarly to Lagrange multipliers.•Rigorous error analysis for optimal order accuracy.•Examples for H(div) approximation, flow exchange across interface, and 3-domain coupling.
This paper presents a family of weak Galerkin finite element methods for elliptic boundary value problems on convex quadrilateral meshes. These new methods use degree
k
≥
0
polynomials separately in ...element interiors and on edges for approximating the primal variable. The discrete weak gradients of these shape functions are established in the local Arbogast–Correa
A
C
k
spaces. These discrete weak gradients are then used to approximate the classical gradient in the variational formulation. These new methods do not use any nonphysical penalty factor but produce optimal-order approximation to the primal variable, flux, normal flux, and divergence of flux. Moreover, these new solvers are locally conservative and offer continuous normal fluxes. Numerical experiments are presented to demonstrate the accuracy of this family of new methods.
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