Robust a Posteriori Error Control and Adaptivity for Multiscale, Multinumerics, and Mortar Coupling

Abstract

We consider discretizations of a model elliptic problem by means of different numerical methods applied separately in different subdomains, termed multinumerics, coupled using the mortar technique. The grids need not match along the interfaces. We are also interested in the multiscale setting, where the subdomains are partitioned by a mesh of size $h$, whereas the interfaces are partitioned by a mesh of much coarser size $H$, and where lower-order polynomials are used in the subdomains and higher-order polynomials are used on the mortar interface mesh. We derive several fully computable a posteriori error estimates which deliver a guaranteed upper bound on the error measured in the energy norm. Our estimates are also locally efficient and one of them is robust with respect to the ratio $H/h$ under an assumption of sufficient regularity of the weak solution. The present approach allows bounding separately and comparing mutually the subdomain and interface errors. A subdomain/interface adaptive refinement s...