An adaptive BDDC method enhanced with prior selected primal constraints

Abstract

An adaptive BDDC (Balancing Domain Decomposition by Constraints) method is considered for three dimensional elliptic problems with coefficients of random variation and high contrast. For such model problems, a certain generalized eigenvalue problem on each subdomain interface is formed to select primal constraints adaptively. In three dimensions, eigenvalue problems are formed on each face or edge nodal equivalence classes. For the case of edges, proposed eigenvalue problems in previous studies are not satisfactory while those for faces perform very effectively. The eigenvalue problems can be enhanced by utilizing prior selected primal constraints. In this paper, this new idea is adopted when forming edge eigenvalue problems and the resulting adaptive BDDC preconditioner is analyzed. In numerical experiments, the new edge eigenvalue problem is also shown to provide a more optimal set of adaptive primal constraints compared to those in the previous studies.

Introduction

An adaptive BDDC (Balancing Domain Decomposition by Constraints) method is developed for three dimensional elliptic problems. For model problems with random and high contrast coefficients, the standard primal unknowns based on the subdomain partition, i.e., unknowns at subdomain vertices, edge averages, and face averages, are not enough and adaptive primal unknowns are included to enhance the coarse component of the BDDC preconditioner. Adaptive primal constraints are selected from well-designed eigenvalue problems on subdomain edges or faces, where edges are nodal equivalence classes shared by more than two subdomains and faces are shared by two subdomains. The adaptive primal unknowns are then formed after change of unknowns on those selected primal constraints. There have been many previous studies to address how to form such eigenvalue problems so as to obtain an effective set of adaptive primal unknowns for the BDDC algorithm for three dimensional problems [1], [2], [3], [4], [5]. We also note that similar approaches have been studied for other types of domain decomposition algorithms, such as overlapping Schwarz methods and FETI(-DP) methods [6], [7], [8], [9], [10], [11], and for BDDC algorithms of isogeometric analysis [12].

For the three dimensional case, a main difficulty in forming such eigenvalue problems is due to the edge nodal equivalence classes, that are shared by more than two subdomains. In the pioneering works [1], [2] by Mandel, Sousedík, and Šístek, eigenvalue problems are formed on the closed faces and the adaptive primal constraints are restricted on open faces discarding values on face boundaries, i.e., edges. Their approach presented promising results for their test models in engineering applications but it was later noticed that the adaptive primal constraints on open faces are not enough to provide a robust preconditioner for some difficult applications [9]. In the works [9], [11] by Klawonn and his coworkers, a more advanced approach was proposed. The adaptive primal constraints chosen from eigenvalue problems on closed faces are applied on both open faces and open edges. In addition, pairwise edge eigenvalue problems are formed to select more adaptive primal constraints, which are needed to obtain a more robust coarse problem in three dimensions. In their work, analysis for estimate of condition numbers was also provided.

In more recent works [3], [4], eigenvalue problems are formed for each open faces and open edges to select the adaptive primal constraints. For the case of open edges, single eigenvalue problem is formed for an edge, where all the subdomains sharing the edge are combined. This resulted a less optimal edge eigenvalue problem and thus the cutting criterion for the eigenvalues when choosing the adaptive primal constraints is not as clear as in the face eigenvalue problem. We note that pairwise edge eigenvalue problems are considered in [11] and for this case a postprocessing is needed to make all the selected primal constraints orthogonal. Estimate of condition numbers was analyzed for the deluxe scaling in [3], and was shown for a general scaling in [4]. In [5], a new idea was proposed to form a more effective eigenvalue problem, where prior selected primal unknowns can be combined when forming eigenvalue problems. Further implementation details and numerical experiments have not been considered in [5].

In this work, by adapting the idea proposed in [5], a new edge eigenvalue problem is formed for a localized finite element space associated to an edge, where previously selected adaptive primal constraints are enforced. In the DD25 meeting [13], the first author also discussed with Clemens Pechstein about some implementation issues for this new approach. Estimate of condition numbers is analyzed for the adaptive BDDC method with the new edge eigenvalue problem and numerical experiments present that the cutting criterion is clear for this new edge eigenvalue problem, just like the face eigenvalue problem, that was shown to perform very effectively for many test examples [3], [4]. In addition, quite a small number of adaptive primal unknowns are chosen from the new edge eigenvalue problem compared to those in [3], [4].

This paper is organized as follows. In Section 2, we introduce a model problem and finite element spaces and in Sections 3 Adaptive BDDC algorithm, 4 Eigenvalue problems and estimate of condition numbers we include a brief description on BDDC algorithms and present how one can form suitable eigenvalue problems on faces and edges to select adaptive primal unknowns. In Section 5, adapting the idea in [5], we propose a new edge eigenvalue problem and provide the estimate of condition numbers for the resulting adaptive BDDC preconditioner. In Section 6, we present numerical results of the proposed method for various test examples to show that the proposed edge eigenvalue problem gives a more effective set of primal unknowns than that in the previous studies.