Mixed Mimetic Spectral Element Method for Stokes Flow: A Pointwise Divergence-Free Solution

TL;DR

This paper introduces a mimetic spectral element method for Stokes flow that ensures pointwise divergence-free solutions by exactly discretizing differential operators, applicable to complex geometries with optimal convergence.

Contribution

The paper develops a higher-order mimetic spectral element method using differential forms for Stokes flow, achieving pointwise divergence-free solutions on curvilinear meshes.

Findings

Method achieves pointwise divergence-free solutions.

Converges optimally on Cartesian and curvilinear meshes.

Validated with multiple test cases.

Abstract

In this paper we apply the recently developed mimetic discretization method to the mixed formulation of the Stokes problem in terms of vorticity, velocity and pressure. The mimetic discretization presented in this paper and in [50] is a higher-order method for curvilinear quadrilaterals and hexahedrals. Fundamental is the underlying structure of oriented geometric objects, the relation between these objects through the boundary operator and how this defines the exterior derivative, representing the grad, curl and div, through the generalized Stokes theorem. The mimetic method presented here uses the language of differential $k$-forms with $k$-cochains as their discrete counterpart, and the relations between them in terms of the mimetic operators: reduction, reconstruction and projection. The reconstruction consists of the recently developed mimetic spectral interpolation functions. The most important result of the mimetic framework is the commutation between differentiation at the continuous level with that on the finite dimensional and discrete level. As a result operators like gradient, curl and divergence are discretized exactly. For Stokes flow, this implies a pointwise divergence-free solution. This is confirmed using a set of test cases on both Cartesian and curvilinear meshes. It will be shown that the method converges optimally for all admissible boundary conditions.