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

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...

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Bibliographic Details
Published in:Journal of computational physics 2018-04, Vol.359, p.312-330
Main Authors: Liu, Jiangguo, Tavener, Simon, Wang, Zhuoran
Format: Article
Language:English
Subjects:
Basis functions
Computational physics
Darcy flow
Finite element analysis
Finite element method
Fluxes
Galerkin method
Hybrid meshes
Linear systems
Lowest order finite elements
Mathematical analysis
Numerical analysis
Parallelograms
Quadrilateral meshes
Quadrilaterals
Triangles
Weak Galerkin
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Summary: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 RT0 for triangles and unmapped RT[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.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2018.01.001