A rigorous Multiscale Method for semi-linear elliptic problems
Please always quote using this URN:urn:nbn:de:0296-matheon-11768
- In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coeffcient functions. For this purpose we construct a generalized finite element basis that spans a low dimensional multiscale space. The basis is assembled by performing localized linear finescale computations in small patches that have a diameter of order H |log(H)| where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coeffcients, we give a rigorous proof for a linear convergence of the H1-error with respect to the coarse mesh size. To solve the arising equations, we propose an algorithm that is based on a damped Newton scheme in the multiscale space.
Author: | Patrick Henning, Axel Malqvist, Daniel Peterseim |
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URN: | urn:nbn:de:0296-matheon-11768 |
Referee: | Ralf Kornhuber |
Document Type: | Preprint, Research Center Matheon |
Language: | English |
Date of first Publication: | 2012/11/27 |
Release Date: | 2012/11/27 |
Tag: | computational homogenization; error analysis; finite element; numerical upscaling |
Institute: | Research Center Matheon |
Humboldt-Universität zu Berlin | |
MSC-Classification: | 35-XX PARTIAL DIFFERENTIAL EQUATIONS / 35Jxx Elliptic equations and systems [See also 58J10, 58J20] / 35J15 Second-order elliptic equations |
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N12 Stability and convergence of numerical methods | |
65-XX NUMERICAL ANALYSIS / 65Nxx Partial differential equations, boundary value problems / 65N30 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods | |
Preprint Number: | 989 |