Transfer of Approximation and Numerical Homogenization of Hyperbolic Boundary Value Problems with a Continuum of Scales
William W. Symes · Rice Research Repository (Rice University) · 2012
Galerkin approximate solutions of two self-adjoint systems with the same right-hand side have errors that mutually dominate each other, provided that the approximating subspaces contain exact solutions to problems with the same right-hand sides for their respective systems. This "transfer of approximation" property was first formulated by Berlyand and Owhadi in a more specialized setting. It provides a very general framework for numerical homogenization, that is, construction of optimal-order finite element approximations for linear elliptic boundary value problems with bounded and measurable coefficients, relying neither for formulation nor proof of convergence on assumed scale separation, periodicity, or ergodicity of coefficients. The construction extends to hyperbolic problems provided that the data is smooth in time. In its basic form, the transfer construction of optimal order approximation has severe practical drawbacks: it produces bases with global support, hence dense stiffness matrices, and at the cost of many solutions of problems as difficult in principle as the original. Numerical experiments suggest that a localization procedure suggested by Owhadi and Zhang still requires basis functions of large support (producing stiffness matrices of high bandwidth) to be effective. The harmonic coordinate construction of Owhadi and Zhang, on the other hand, can be viewed as a transfer-of-approximation instance, and produces stiffness matrices with the same sparsity pattern as standard conforming finite element methods.