Weak Approximation Properties of Elliptic Projections with Functional Constraints

Abstract

This paper is on the construction of energy-minimizing coarse spaces that obey certain functional constraints and can thus be used, for example, to build robust coarse spaces for elliptic problems with large variations in the coefficients. In practice they are built by patching together solutions to appropriate local saddle point or eigenvalue problems. We develop an abstract framework for such constructions, akin to an abstract Bramble–Hilbert-type lemma, and then apply it in the design of coarse spaces for discretizations of PDEs with highly varying coefficients. The stability and approximation bounds of the constructed interpolant are in the weighted norm and are independent of the variations in the coefficients. Such spaces can be used, for example, in two-level overlapping Schwarz algorithms for elliptic PDEs with large coefficient jumps generally not resolved by a standard coarse grid or for numerical upscaling purposes. Some numerical illustration is provided.