Iterative Solvers Based on Domain Decomposition

Abstract

The numerical approximation of partial differential equations, very often, is a challenging task. Many such problems of practical interest can only be solved by means of modern supercomputers. However, the efficiency of the simula- tion depends strongly on the use of special numerical algorithms. Domain de- composition methods provide powerful tools for the numerical approximation of partial differential equations arising in the modeling of many interesting applicat ions in science and engineering. Although the first domain decompo- sition techniques were used successfully more than hundred years ago, these methods are relatively new for the numerical approximation of partial differ- ential equations. The possibilities of high performance computations and the interest in large-scale problems have led to an increased research activity in the field of domain decomposition. However, the meaning of the term "domain decomposition" depends strongly on the context. It can refer to optimal discretiz ation techniques for the underlying problems, or to efficient iterative solvers for the arising large systems of equations, or to parallelization techniques. In many modern simulation codes, different aspects of domain decomposition techniques come into play, and the overall efficiency depends on a smooth interaction between these different components. The coupling of different discretization schemes , the coupling of different physical models, and many efficient preconditioners for the algebraic systems can be analyzed within an abstract framework. At first glance these aspects seem to be rather independent. However, all have one central idea in common: The decomposition of the underlying global problem into suitable subproblems of smaller complexity. In general, a com- plete decoupling of the global problem into many independent subproblems, which are easy to solve, is not possible. Since, the subproblems are very often coupled, there has to be communication between the different subproblems. Although the term optimal depends on the context, the proper handling of the inform at ion transfer across the interfaces between the subproblems is of major importance for the design of opt imal methods. In the case of dis- cretization techniques, a priori estimates for the discretization errors have to be considered. They very much depend on the appro priate couplings across the interfaces which are often realized by matching conditions. The jump across the int erfaces which measures the nonconformity of the method has to be bounded in a suitable way. In the case of iterative solvers , the convergence rate and the computational effort for one iteration step measure the quality of a method. To obtain scalable iteration schemes, very often, one has to includ e a suitable global problem of small complexity. In this work, both discretiz ation techniques and iterative solvers are ad- dressed. A brief overview of different approaches is given and new techniques and ideas are proposed. An abst ract framework for domain decomposition methods is presented and an analysis is carried out for new techniques of spe- cial interest . Optimal est imates for the methods considered are established and numerical results confirm the theoreti cal predictions. Chapter 1 concerns special discretiz ation methods based on domain de- composition techniques. In particular , the decomp osition of geomet rical com- plex structures into sub domains of simple shape is of special int erest . Another example is the decomposition into substructures on which different physical models are relevant . Then, for each of these subproblems, an optimal ap- proximation scheme involving the choice of the triangulation as well as the discretization can be chosen. However to obtain optimal discretizations for the global problem, the discrete subproblems have been glued together ap- propriately. Here, we focus on mortar finite element methods. To start, we review the standard mort ar setting for the coupling of La- grangian conforming finite elements in Sect . 1.1. Both standard mortar for- mulations - the nonconforming positive definite problem and the saddle point problem based on the unconstrained product space - are given. In Sect . 1.2, we introduce and analyze alternative Lagrange multiplier spaces. We derive abstract conditions on the Lagrange multiplier spaces such that the non conforming discretization schemes obtained yield optimal a pri- ori results. Lagrange multiplier spaces based on a dual basis are of special int erest . In such a case, a biorthogonality relation between the nodal basis functions of these spaces and th e finite element trace spaces holds . A main advantage of these new Lagrange multiplier spaces is that the locality of the support of the nodal basis functions of the constrained space can be pr eserved. With this observation in mind, we introduce a new equivalent mortar formulation defined on the unconst rained product space in Sect . 1.3. We show that the non-symmetric formulation can be analyzed as a Dirichlet - Neumann coupling. Based on the elimination of the Lagrange multiplier , we derive a symmetric positive definite formul at ion on the unconstrained product space, and the equivalence to th e positive definite problem on th e constrained space is shown. Two formulations, a variational as well as an algebraic one, are presented and discussed. A standard nodal basis for th e unconstrained product space can be used in the implement at ion. The stiffness matrix associated with our new variational form can be obtained from the standard one on the unconstrained space by local operations. Section 1.4 concerns two examples of non-standard mortar situations. Each of them reflects an int eresting feature of the abstract general frame- work, and illustrates the flexibility of the method. We start with the cou- pling of two different discretization schemes. The matching at the int erface is based on the dual role of Dirichlet and Neumann boundary conditions. Two different equivalent formulations are given for the coupling of mixed and standard conforming finite elements . In our second example, we rewrite the nonconforming Crouzeix- Raviart finite elements as mortar finite elements. We consider the extreme case that the decomposition of th e domain is given by the fine triangulation and that therefore the number of subdomains tends to infinity as the discretization paramet er of the t riangulat ion tends to zero . Finally in Sect . 1.5, we present several series of numerical results. In par- ticular, we study the influence of the choice of the Lagrange multiplier space on the discretization erro rs . Examples with severa l crosspoints, a corne r sin- gularity, discontinuous coefficients, a rotating geomet ry, and a linear elast ic- ity problem are considered. A second test series concerns th e influence of the choice of the non-mortar side. Adaptive and uniform refinement techniques are applied. In our last te st series, we consider the influence of jumps in th e coefficient on an adapt ive refinement process at the interface. Chapter 2 concerns iterative solut ion techniques based on domain de- composit ion. A brief overview of general Schwarz meth ods, including multi- grid techniques, is given in Sect. 2.1. Examples for the standard H I-case illust rate overlapping, non-overlapping, and hierar chical decomposition tech- niqu es. The following sect ions contain new results on non-stand ard situat ions; we discuss vector field discretizations as well as mortar methods. Section 2.2 focuses on an iterative subst ructuring and a hierarchical basis method for Raviart-Thomas finite elements in 3D. We start with the defini- tion of th e local spaces and th e relevant bilinear forms and subspaces. The central result of this section is established in Subsect. 2.2.2; it is a poly- logarithmical bound independent of the jumps of the coefficients across the subdomain boundaries of our iterat ive substructur ing method. The technical tools are discussed in det ail with particular emphasis on the role of trace the- orems, harmonic extensions, and dual norms applied to finite element spaces . As in the 2D case for standard Lagrangian finite elements , we introduce three different types of subspaces called VII , Vp , and VT . We cannot avoid the use of a global space to obtain quasi-optimal bounds. But in contrast to th e stan- dard Lagrangi an finite elements in 3D, the low dimensional Raviar t - Thomas space associated with the macro-triangulation formed by th e subregions can be used to obtain quasi-optimal results where the constant does not depend on the jumps of the coefficients across the subdomain boundaries. Sections 2.3-2.5 concern different iterative solvers for mortar finite ele- ment formulations. In Sect. 2.3, we combine the idea of dual basis functions for the Lagrange multiplier space with standard multigrid techniques for sym- metric positive definite systems. The new mortar formulation, analyzed in Sect. 1.3, is the point of departure for the introduction of our iterative solver. We define and analyze our multigrid method in terms of level dependent bilin- ear forms, modified transfer operators, and a special class of smoothers which includes a standard Gaufl-Seidel smoother. Convergence rates independent of the number of refinement steps are established for the W-cycle provided that the number of smoothing steps is large enough. The numerical results confirm the theory. Moreover asymptotically constant convergence rates are obtained for the V-cycle with one pre- and one postsmoothing step. Section 2.4 concerns a Dirichlet-Neumann type algorithm for the mortar method. It turns out to be a block Gaufi-Seidel solver for the unsymmetric mortar formulation on the product space. Numerical results illustrate the influence of the choice of the damping parameter. The transfer of the bound- ary values at the interface is realized in terms of a scaled mas