Primal interface formulation for coupling multiple PDEs: A consistent derivation via the Variational Multiscale method

Abstract

This paper presents a primal interface formulation that is derived in a systematic manner from a Lagrange multiplier method to provide a consistent framework to couple different partial differential equations (PDE) as well as to tie together nonconforming meshes. The derivation relies crucially on concepts from the Variational Multiscale (VMS) approach wherein an additive multiscale decomposition is applied to the primary solution field. Modeling the fine scales locally at the interface using bubble functions, consistent residual-based terms on the boundary are obtained that are subsequently embedded into the coarse-scale problem. The resulting stabilized Lagrange multiplier formulation is converted into a robust Discontinuous Galerkin (DG) method by employing a discontinuous interpolation of the multipliers along the segments of the interface. As a byproduct, analytical expressions are derived for the stabilizing terms and weighted numerical flux that reflect the jump in material properties, governing equation, or element geometry across the interface. Also, a procedure is proposed for automatically generating the fine-scale bubble functions that is inspired by a performance study of residual-free bubbles for the interface problem. A series of numerical tests confirms the robustness of the method for solving interface problems with heterogeneous elements, materials, and/or governing equations and also highlights the benefit and importance of deriving the flux and stabilization terms.