Numerical homogenization tries to approximate the solutions of elliptic partial differential equations with strongly oscillating coefficients by functions from modified finite element spaces. We present a class of such methods that are closely related to the methods that have recently been proposed by Målqvist and Peterseim [Math. Comp. 83, 2014, pp. 2583-2603. Like these methods, the new methods do not make explicit or implicit use of a scale separation. Their comparatively simple analysis is based on the theory of additive Schwarz or subspace decomposition methods.
Abstract. Let u e H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some S, S being a suitable finite-element space. Efficient and reliable a posteriod estimates of the error u II, measuring the (local) quality of , play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well-known class of error estimates can be derived systematically by localizing the discretized defect problem by using domain decomposition techniques. In this paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.
Numerical homogenization tries to approximate solutions of elliptic partial differential equations with strongly oscillating coefficients by the solution of localized problems over small subregions. We develop and analyze a rapidly convergent iterative method for numerical homogenization that shares this feature with existing approaches and is modeled after the Schwarz method. The method is highly parallelizable and of lower computational complexity than comparable methods that as ours do not make explicit or implicit use of a scale separation.
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