Parallel finite cell method with adaptive geometric multigrid

Abstract

Generation of appropriate computational meshes in the context of numerical methods for partial differential equations is technical and laborious and has motivated a class of advanced discretization methods commonly referred to as unfitted finite element methods. To this end, the finite cell method (FCM) combines high-order FEM, adaptive quadrature integration and weak imposition of boundary conditions to embed a physical domain into a structured background mesh. While unfortunate cut configurations in unfitted finite element methods lead to severely ill-conditioned system matrices that pose challenges to iterative solvers, such methods permit the use of optimized algorithms and data patterns in order to obtain a scalable implementation. In this work, we employ linear octrees for handling the finite cell discretization that allow for parallel scalability, adaptive refinement and efficient computation on the commonly regular background grid. We present a parallel adaptive geometric multigrid with Schwarz smoothers for the solution of the resultant system of the Laplace operator. We focus on exploiting the hierarchical nature of space tree data structures for the generation of the required multigrid spaces and discuss the scalable and robust extension of the methods across process interfaces. We present both the weak and strong scaling of our implementation up to more than a billion degrees of freedom on distributed-memory clusters.