Abstract
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.
Highlights
The multimesh finite element method presented in [1] extends the finite element method to arbitrarily many overlapping and intersecting meshes
Nitsche’s method is the basis for discontinuous Galerkin methods [5] which may be cast in a setting of non-matching meshes [6,7,8,9,10]
We have analyzed a general framework for discretization of the Poisson equation posed on a domain defined by an arbitrary number of intersecting meshes with arbitrary mesh sizes
Summary
The multimesh finite element method presented in [1] extends the finite element method to arbitrarily many overlapping and intersecting meshes This is of great value for problems that are naturally formulated on domains composed of parts, such as complex domains composed of simpler parts that may be more meshed than their composition. The analysis holds for two and three dimension as well as for higher order elements, and extends previous works on cut finite elements for overlapping meshes and interface problems to much more general mesh arrangements and mesh sizes. In the remainder of this paper, we analyze the multimesh finite element method for the Poisson problem for an arbitrary number of intersecting meshes and arbitrarily mesh sizes, and present numerical examples.
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