Fast-convergent two-level restricted additive Schwarz methods based on optimal local approximation spaces

TL;DR

This paper rigorously proves that the two-level restricted additive Schwarz (RAS) method for multiscale PDEs, built on top of a multiscale spectral generalized finite element method (MS-GFEM), converges at a rate of $\Lambda$, where $\Lambda$ represents the error of the underlying MS-GFEM.

Abstract

This paper proposes a two-level restricted additive Schwarz (RAS) method for multiscale PDEs, built on top of a multiscale spectral generalized finite element method (MS-GFEM). The method uses coarse spaces constructed from optimal local approximation spaces, which are based on local eigenproblems posed on (discrete) harmonic spaces. We rigorously prove that the method, used as an iterative solver or as a preconditioner for GMRES, converges at a rate of $Λ$, where $Λ$ represents the error of the underlying MS-GFEM. The exponential convergence property of MS-GFEM, which is indepdendent of the fine mesh size $h$ even for highly oscillatory and high contrast coefficients, thus guarantees convergence in a few iterations with a small coarse space. We develop the theory in an abstract framework, and demonstrate its generality by applying it to various elliptic problems with highly heterogeneous coefficients, including $H({\rm curl})$ elliptic problems. The performance of the proposed method is systematically evaluated and illustrated via applications to two and three dimensional heterogeneous PDEs, including challenging elasticity problems in realistic composite aero-structures.

Figures (8)

• Figure 1: Diffusion equation example. Left: 'skyscraper' coefficient; right: solution.
• Figure 2: Results for the diffusion example solved with MS-GFEM as the iterative method. We report the iteration count (left) and total computational time in seconds (right) for different numbers of oversampling layers and local eigenfunctions used per subdomain. The numbers in brackets on the right show the time spent on the local eigensolves.
• Figure 3: Results for the diffusion example solved with MS-GFEM as the iterative method. The plots show the residual reduction over iterations (left) and over time (right) for selected parameters used in \ref{['heatmaps']}.
• Figure 4: Results for the diffusion example solved with different preconditioner schemes. The plots show the convergence history of these algorithms with the MS-GFEM coarse space (left) and the GenEO coarse space (right), all with a fixed choice of oversampling and local space sizes (Ovsp=8, #Eig=24).
• Figure 5: Numerical results for the diffusion example. The top plots show the convergence rates of different MS-GFEM based solvers for increasing oversampling layers (left, with #Eig = 24 fixed) and numbers of local bases (right, with Ovsp=8 fixed). The bottom plots show the required computational time of the iterative MS-GFEM method as functions of the numbers of oversampling layers (left, with #Eig = 12 fixed) and local bases (right, with Ovsp=8 fixed).

Theorems & Definitions (28)

• Theorem 2.1
• Lemma 2.2
• Corollary 2.3
• Theorem 2.4
• Remark 2.5
• Definition 3.1
• Lemma 3.2
• Definition 3.3: Iterative MS-GFEM
• Proposition 3.4
• Proof 1