An analysis of a class of variational multiscale methods based on subspace decomposition
For researchers in numerical homogenization, this work offers a simplified theoretical framework for a known class of methods, but it is incremental as it builds directly on existing work.
The paper presents a class of numerical homogenization methods for elliptic PDEs with oscillating coefficients, closely related to the Målqvist–Peterseim method, and provides a simplified analysis based on variational multiscale and subspace decomposition methods.
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 in this paper a class of such methods that are very closely related to the method of Målqvist and Peterseim [Math. Comp. 83, 2014]. Like the method of Målqvist and Peterseim, these methods do not make explicit or implicit use of a scale separation. Their compared to that in the work of Målqvist and Peterseim strongly simplified analysis is based on a reformulation of their method in terms of variational multiscale methods and on the theory of iterative methods, more precisely, of additive Schwarz or subspace decomposition methods.