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

We present a novel method for solving high-order partial differential equations (PDEs) over planar multi-patch geometries with possibly extraordinary vertices demonstrated on the basis of the polyharmonic equation of order m, m ≥ 1, which is a particular linear elliptic PDE of order 2 m . Our approach is based on the concept of Isogeometric Tearing and Interconnecting (IETI) and allows to couple the numerical solution of the PDE with C s -smoothness, s ≥ 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, 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 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 for the polyharmonic equation of order m = 1 , m = 2 and m = 3 , i.e. for the Poisson’s, the biharmonic and the triharmonic equation, respectively, are shown to demonstrate the potential of our IETI method for solving high-order problems over planar multi-patch geometries with possibly extraordinary vertices.