Hybridized weak Galerkin finite element methods for Brinkman equations. (English) Zbl 1476.65304
Summary: This paper presents a hybridized weak Galerkin (HWG) finite element method for solving the Brinkman equations. Mathematically, Brinkman equations can model the Stokes and Darcy flows in a unified framework so as to describe the fluid motion in porous media with fractures. Numerical schemes for Brinkman equations, therefore, must be designed to tackle Stokes and Darcy flows at the same time. We demonstrate that HWG is capable of providing very accurate and stable numerical approximations for both Darcy and Stokes. The main features of HWG is that it approximates the differential operators by their weak forms as distributions and it introduces the Lagrange multipliers to relax certain constraints. We establish the optimal order error estimates for HWG solutions of Brinkman equations. We also present a Schur complement formulation of HWG, which reduces the systems’ computational complexity significantly. A number of numerical experiments are provided to confirm the theoretical developments.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35J50 | Variational methods for elliptic systems |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 76S05 | Flows in porous media; filtration; seepage |
| 35Q35 | PDEs in connection with fluid mechanics |
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