Overlapping Schwarz preconditioned eigensolvers for spectral element discretizations

Abstract

Model generalized eigenproblems associated with self-adjoint differential operators in nonstandard homogeneous or heterogeneous domains are considered. Their numerical approximation is based on Gauss–Lobatto–Legendre conforming spectral elements defined by Gordon–Hall transfinite mappings. The resulting discrete eigenproblems are solved iteratively with a Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method, accelerated by an overlapping Schwarz preconditioner. Several numerical tests show the good convergence properties of the proposed preconditioned eigensolver, such as its scalability and quasi-optimality in the discretization parameters, which are analogous to those obtained for overlapping Schwarz preconditioners for linear systems.

Introduction

Many interesting scientific applications require the accurate evaluation of the smallest eigenpairs of large sparse matrices. In recent years, several iterative methods have been proposed for this task, trying to extend to eigenproblems the promising results obtained by the research community on preconditioned iterative methods for linear systems, see e.g. [22, Ch. 16] and the references therein.

In this paper, we consider a model generalized eigenproblem $A\mathbf{u}=\lambda B\mathbf{u}$ associated with a self-adjoint differential operator, in nonstandard homogeneous or heterogeneous domains. The problem is discretized by the standard conforming Spectral Element Method (SEM) based on quadrilateral elements and Gauss–Lobatto–Legendre (GLL) quadrature points, so the method can be viewed as a nodal version of hp Finite Element Methods (FEM); see, e.g., [5], [2], [8], [17], [29]. One difficulty in the implementation of the SEM is the approximation of problems in complex-shaped domains, arising in several branches of applied sciences. We shall address this point by using transfinite interpolation, or Gordon–Hall maps [15], in order to build very flexible maps from the reference square domain to a generic spectral element with quadrilateral shape and, in the general case, with curvilinear edges. In our previous work [16] on preconditioned iterative solvers for SEM–GLL linear systems, we showed that the good convergence properties (with respect to the discretization parameters h and p) of Schwarz preconditioners for standard rectangular elements are retained for SEM–GLL elements with Gordon–Hall maps. This previous study is here extended to eigenproblems arising from SEM–GLL discretizations.

In the FEM framework, several large mesh eigenproblems arising from mathematical physics have been addressed by means of preconditioned eigensolvers; see, e.g. [1], [3], [4], [12], [9], [21], [7], [20], [22]. Here, we consider the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method [20], [19] that has been proposed for the numerical solution of large-scale, generalized symmetric positive definite eigenvalue problems. The SEM discrete systems arising at each LOBPCG iteration are preconditioned by the overlapping Schwarz (OS) method. The latter is based on partitioning the domain of the given problem into overlapping subdomains, solving parallel independent local problems on the subdomains, and solving a coarse problem on a coarse mesh, in order to ensure scalability; see, e.g., [11], [32] for a general introduction to OS methods, and [6], [13], [14], [16], [25] for OS applications to SEM discretizations.

The outline of this paper is as follows. We introduce the model eigenvalue problem and its SEM approximation in Section 2. In order to deal with nonstandard geometries, Gordon–Hall transfinite interpolation is introduced in Section 3. In Section 4, we recall a family of preconditioned, iterative eigensolvers including the LOBPCG method. In Section 5, the classical domain decomposition overlapping Schwarz preconditioner is applied to the LOBPCG eigensolver. The paper is concluded, in Section 6, by several numerical test problems showing the convergence properties of the LOBPCG-OS preconditioner with respect to the discretization parameters $H\text{,}h\text{,}p$.