A Localized Orthogonal Decomposition Method for Heterogeneous Stokes Problems

Abstract

In this paper, we propose a multiscale method for heterogeneous Stokes problems. The method is based on the Localized Orthogonal Decomposition (LOD) methodology and has approximation properties independent of the regularity of the coefficients. We apply the LOD to an appropriate reformulation of the Stokes problem, which allows us to construct exponentially decaying basis functions for the velocity approximation while using a piecewise constant pressure approximation. The exponential decay motivates a localization of the basis computation, which is essential for the practical realization of the method. We perform a rigorous a priori error analysis and prove optimal convergence rates for the velocity approximation and a post-processed pressure approximation, provided that the supports of the basis functions are logarithmically increased with the desired accuracy. Numerical experiments support the theoretical results of this paper.

Figures (4)

• Figure 6.1: Initial mesh for the mesh generation (left). Multiscale coefficient used in all numerical experiments (right).
• Figure 6.2: Decay of the modulus of a prototypical LOD basis function, plotted using a logarithmic color scale (left). $H^1$-errors of the localized approximation of the prototypical LOD basis functions for several coarse mesh sizes $H$, plotted as a function of the oversampling parameter $\ell$ (right).
• Figure 6.3: $L^2$-errors (left) and $H^1$-errors (right) for the velocity approximation for several oversampling parameters $\ell$, plotted as functions of the coarse mesh size $H$.
• Figure 6.4: $L^2$-errors of the post-processed pressure approximation for several oversampling parameters $\ell$, plotted as functions of the coarse mesh size $H$ (left). $L^2$-errors of the pressure approximation computed with respect to $\Pi_H p$ for several coarse mesh sizes $H$, plotted as a function of the oversampling parameter $\ell$ (right).

Theorems & Definitions (15)

• Lemma 3.1: Prototypical basis
• proof : Proof of \ref{['le:protbasis']}
• Theorem 3.2: Prototypical method
• proof
• Theorem 4.1: Exponential decay
• proof
• Theorem 4.2: Localization error
• proof
• Theorem 5.1: Localized method
• proof