Multiscale techniques for parabolic equations
نویسندگان
چکیده
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 [Formula: see text]-norm. We present numerical examples, which confirm our theoretical findings.
منابع مشابه
Multiscale Approach to Parabolic Equations Derivation: Beyond the Linear Theory
The concept of the iterative parabolic approximation based on the multiscale technique is discussed. This approach is compared with the traditional ways to derive the wide-angle parabolic equation. While the latter fail in the nonlinear case, the multiscale derivation technique leading to iterative parabolic equations can be easily adapted to handle it. The nonlinear iterative parabolic approxi...
متن کاملMultiscale Analysis and Computation for Parabolic Equations with Rapidly Oscillating Coefficients in General Domains
Abstract. This paper presents the multiscale analysis and computation for parabolic equations with rapidly oscillating coefficients in general domains. The major contributions of this study are twofold. First, we define the boundary layer solution and the convergence rate with ε1/2 for the multiscale asymptotic solutions in general domains. Secondly, a highly accurate computational algorithm is...
متن کاملA Stable and Accurate Explicit Scheme for Parabolic Evolution Equations
We show that the combination of several numerical techniques, including multiscale preconditionning and Richardson extrapolation, yields stable and accurate explicit schemes with large time steps for parabolic evolution equations. Our theoretical study is limited here to linear problems (typically the Heat equation with non-constant coee-cients). However, the extrapolation procedures that we st...
متن کاملMultiscale problems and homogenization for second-order Hamilton–Jacobi equations
We prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic PDEs. As a special case we cover the iterated homogenization for such equations with oscillating initial data. Explicit examples, among others, are the two-scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n-sca...
متن کاملEfficient algorithms for multiscale modeling in porous media
We describe multiscale mortar mixed finite element discretizations for second order elliptic and nonlinear parabolic equations modeling Darcy flow in porous media. The continuity of flux is imposed via a mortar finite element space on a coarse grid scale, while the equations in the coarse elements (or subdomains) are discretized on a fine grid scale. We discuss the construction of multiscale mo...
متن کامل