Multiscale hybrid-mixed method for the Stokes and Brinkman equations—The method

Abstract

The multiscale hybrid-mixed (MHM) method is extended to the Stokes and Brinkman equations with highly heterogeneous coefficients. The approach is constructive. We first propose an equivalent dual-hybrid formulation of the original problem using a coarse partition of the heterogeneous domain. Faces may be not aligned with jumps in the data. Then, the exact velocity and the pressure are characterized as the solution of a global face problem and the solutions of local independent Stokes (or Brinkman) problems at the continuous level. Owing to this decomposition, the one-level MHM method stems from the standard Galerkin approach for the Lagrange multiplier space. Basis functions are responsible for upscaling the unresolved scales of the medium into the global formulation. They are the exact solution of the local problems with prescribed Neumann boundary conditions on faces driven by the Lagrange multipliers. We make the MHM method effective by adopting the unusual stabilized finite element method to solve the local problems approximately. As such, equal-order interpolation turns out to be an option for the velocity, the pressure and the Lagrange multipliers. The numerical solutions share the important properties of the continuum, such as local equilibrium with respect to external forces and the local mass conservation. Several academic and highly heterogeneous tests infer that the method achieves super-convergence for the velocity as well as optimal convergence for the pressure and also for the stress tensor in their natural norms.

A new staggered DG method for the Brinkman problem robust in the Darcy and Stokes limits

“…The use of Stokes-tailored is analyzed in [34], where a new nonconforming element is constructed. In addition to the aforementioned works, one can also refer to [10,11,39,4,8,26,32,22,24,38,2,1,6,28,3,23] and the references therein for a glance at uniformly stable methods for the Brinkman problem. Recently, some novel methods have been proposed to solve the Brinkman problem on general quadrilateral and polygonal meshes, which is tricky and usually requires special treatment in order to deliver robust results with respect to the rough grids.…”

The multiscale hybrid mixed method in general polygonal meshes

“…The global problem needs for its construction the solution of local problems that fulfill the role of upscaling the under-mesh structures. Introduced and analysed in [24,2,31] for the Laplace (Darcy) equation, the MHM method has been further extended to other elliptic problems in [25,23] as well as to mixed and hyperbolic models in [3] and [29], respectively. See also [26] for an abstract setting for the MHM method.…”

On a multiscalea posteriorierror estimator for the stokes and Brinkman equations

“…This work proposes a new residual a posteriori error estimator for the MHM method applied to the Stokes/Brinkman problem. It accounts for the multi-level numerical approximation of the method presented in [7]. Indeed, the error indicator has two-levels of contributions: η 1 related to the jump of the discrete velocity on the skeleton of the first-level mesh, similar to those that appear in previous works (see [6,24], for instance), and η 2 coming from the approximation error of the solutions at the local problems.…”

“…The paper is organized as follows: Section 2 introduces the model problem, and Section 3 revisits the one and two-level MHM methods proposed in [7], and gives some preliminary results. Section 4 presents the multiscale a posteriori error estimator and its analysis.…”