[Paper Review] A rigorous Multiscale Method for semi-linear elliptic problems

Patrick Henning, Axel Målqvist+1 more|arXiv (Cornell University)|Nov 27, 2012

Advanced Mathematical Modeling in Engineering28 references3 citations

TL;DR

This paper presents a rigorous multiscale method for solving semi-linear elliptic problems with highly oscillatory, heterogeneous coefficients. It constructs a generalized finite element basis via localized fine-scale computations in patches of size H log(H⁻¹), enabling linear H¹-convergence with respect to the coarse mesh size H, and employs a damped Newton scheme for solving the resulting multiscale system.

ABSTRACT

In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problemswith heterogeneous andhighly variable coefficient functions. For this purposewe construct a generalized finiteelement basisthat spansalow dimensionalmultiscale space. The basis is assembled by performing localized linear fine-scale computations in small patches that have a diameter of order Hlog(H −1) where H is the coarse mesh size. Without any assumptions on the type of the oscillations in the coefficients, we give a rigorous proof for a linear convergence of the H 1-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.

Motivation & Objective

• To develop a robust multiscale method for semi-linear elliptic problems with highly variable and heterogeneous coefficients.
• To construct a low-dimensional multiscale space that accurately captures fine-scale features without restrictive assumptions on coefficient oscillations.
• To ensure convergence of the method with a rigorous error analysis under minimal regularity assumptions.
• To design an efficient solution algorithm for the resulting multiscale system using a damped Newton scheme.

Proposed method

• Construct a generalized finite element basis by solving local fine-scale problems in patches of diameter H log(H⁻¹), where H is the coarse mesh size.
• Assemble the multiscale space by combining local solutions that span the fine-scale behavior of the heterogeneous coefficients.
• Formulate the global multiscale problem in the reduced-dimensional multiscale space, preserving the essential features of the original problem.
• Apply a damped Newton scheme to solve the nonlinear system arising from the Galerkin discretization in the multiscale space.
• Ensure convergence of the iterative solver by controlling step lengths to maintain global convergence properties.
• Perform rigorous error analysis to establish linear convergence of the H¹-error with respect to the coarse mesh size H.

Experimental results

Research questions

• RQ1 Can a multiscale method be constructed for semi-linear elliptic problems with highly oscillatory coefficients without assuming periodicity or scale separation?
• RQ2 What is the optimal patch size for local computations that balances accuracy and computational efficiency in the multiscale basis construction?
• RQ3 Does the proposed method achieve linear convergence in the H¹-norm with respect to the coarse mesh size H?
• RQ4 How can a robust and efficient iterative solver be designed for the resulting nonlinear multiscale system?
• RQ5 Can the convergence analysis be rigorously established without restrictive assumptions on the coefficient variability?

Key findings

• The method achieves linear convergence of the H¹-error with respect to the coarse mesh size H, independent of coefficient oscillation type.
• The multiscale basis is constructed via localized computations in patches of size H log(H⁻¹), ensuring computational efficiency and scalability.
• The method does not require assumptions on the nature of coefficient oscillations, such as periodicity or scale separation.
• The damped Newton scheme ensures robust convergence for the nonlinear system arising in the multiscale formulation.
• The resulting multiscale space is low-dimensional and captures the essential fine-scale features necessary for accurate solution approximation.
• The convergence analysis is rigorous and holds under minimal assumptions on the coefficient functions.