An Isogeometric Tearing and Interconnecting (IETI) method for solving high order partial differential equations over planar multi-patch geometries

Abstract

We present a novel method for solving high-order partial differential equations (PDEs) over planar multi-patch geometries demonstrated on the basis of the polyharmonic equation of order $m$, $m \geq 1$, which is a particular linear elliptic PDE of order $2m$. Our approach is based on the concept of Isogeometric Tearing and Interconnecting (IETI) [43] and allows to couple the numerical solution of the PDE with $C^s$-smoothness, $s \geq m-1$, across the edges of the multi-patch geometry. The proposed technique relies on the use of a particular class of multi-patch geometries, called bilinear-like $G^s$ multi-patch parameterizations [37], to represent the multi-patch domain. The coupling between the neighboring patches is done via the use of Lagrange multipliers and leads to a saddle point problem, which can be solved efficiently first by a small dual problem for a subset of the Lagrange multipliers followed by local, parallelizable problems on the single patches for the coefficients of the numerical solution. Several numerical examples of solving the polyharmonic equation of order $m=2$ and $m=3$, i.e. the biharmonic and triharmonic equation, respectively, over different multi-patch geometries are shown to demonstrate the potential of our IETI method for high-order problems.