Multimesh finite elements with flexible mesh sizes

We analyze a new framework for expressing finite element methods on arbitrarily many intersecting meshes: multimesh finite element methods. The multimesh finite element method, first presented in Johansson et al. (2019), enables the use of separate meshes to discretize parts of a computational domain that are naturally separate; such as the components of an engine, the domains of a multiphysics problem, or solid bodies interacting under the influence of forces from surrounding fluids or other physical fields. Furthermore, each of these meshes may have its own mesh parameter. In the present paper we study the Poisson equation and show that the proposed formulation is stable without assumptions on the relative sizes of the mesh parameters. In particular, we prove optimal order a priori error estimates as well as optimal order estimates of the condition number. Throughout the analysis, we trace the dependence of the number of intersecting meshes. Numerical examples are included to illustrate the stability of the method.