Abstract
We use the local orthogonal decomposition technique introduced in Målqvist and Peterseim (Math Comput 83(290):2583–2603, 2014) to derive a generalized finite element method for linear and semilinear parabolic equations with spatial multiscale coefficients. We consider nonsmooth initial data and a backward Euler scheme for the temporal discretization. Optimal order convergence rate, depending only on the contrast, but not on the variations of the coefficients, is proven in the L_infty (L_2)-norm. We present numerical examples, which confirm our theoretical findings.
Highlights
In this paper we study numerical solutions to parabolic equations with highly varying coefficients
Using tools from classical finite element theory for parabolic equations, see, e.g, [11,12,18], and references therein, we prove convergence of optimal order in the L∞(L2)-norm for linear and semilinear equations under minimal regularity assumptions and nonsmooth initial data
The purpose of the method described in this paper is to find an approximate solution, let us denote it by Ufor in some space V ⊂ Vh, such that dim V = dim VH, for H > h, and the error Un−Un ≤ C H 2
Summary
In this paper we study numerical solutions to parabolic equations with highly varying coefficients. The method in [13], often referred to as local orthogonal decomposition, constructs a generalized finite element space where the basis functions contain information from the diffusion coefficient and have support on small vertex patches. With this approach, convergence of optimal order can be proved for an arbitrary positive and bounded diffusion coefficient. Using tools from classical finite element theory for parabolic equations, see, e.g, [11,12,18], and references therein, we prove convergence of optimal order in the L∞(L2)-norm for linear and semilinear equations under minimal regularity assumptions and nonsmooth initial data.
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