Coupling non-conforming discretizations of PDEs by spectral approximation of the Lagrange multiplier space

TL;DR

This paper introduces a spectral approximation-based non-conforming domain decomposition method for PDEs, enabling flexible coupling of different discretizations and meshes through Lagrange multipliers on interfaces.

Contribution

It proposes a novel spectral approach to discretize interface Lagrange multipliers independently of subdomain discretizations, enhancing coupling flexibility for various numerical methods.

Findings

Effective for non-conforming meshes with different polynomial degrees

Suitable for coupling finite element, spectral element, and isogeometric methods

Demonstrated robustness in numerical simulations

Abstract

This work focuses on the development of a non-conforming domain decomposition method for the approximation of PDEs based on weakly imposed transmission conditions: the continuity of the global solution is enforced by a discrete number of Lagrange multipliers defined over the interfaces of adjacent subdomains. The method falls into the class of primal hybrid methods, which also include the well-known mortar method. Differently from the mortar method, we discretize the space of basis functions on the interface by spectral approximation independently of the discretization of the two adjacent domains; one of the possible choices is to approximate the interface variational space by Fourier basis functions. As we show in the numerical simulations, our approach is well-suited for the solution of problems with non-conforming meshes or with finite element basis functions with different polynomial degrees in each subdomain. Another application of the method that still needs to be investigated is the coupling of solutions obtained from otherwise incompatible methods, such as the finite element method, the spectral element method or isogeometric analysis.