Algebraic Multigrid Methods for Vector-Field Problems

This project is concerned with Algebraic Multigrid (AMG) methods for the solution of large-scale systems of linear algebraic equations arising from finite element (FE) discretization of (systems of) elliptic partial differential equations (PDEs). In particular, we address differential operators with a large (near) nullspace. Borderline cases of hyperbolic problems with complex characteristics will have to be considered as well. Our general objectives are the design, analysis and implementation of new AMG and Algebraic Multilevel (AML) preconditioners that enable an efficient solution of direct field problems in this category: the main emphasis is on problems arising from the discretization of Maxwell`s equations, solid and structural mechanical problems with bad parameters, and problems arising in computational fluid dynamics. The research plan comprises the following components: 1. Investigation of element-based AMG and AML methods regarding non-conforming FE and Discontinuous Galerkin (DG) discretizations. 2. Development of element-, face-, and edge-based strategies for the generation of adequate coarse-grid problems. 3. AMG for non-symmetric and indefinite matrices: Application to (scalar) convection- diffusion, Stokes, and Oseen equations. 4. AMG for non-M matrices: Application to Maxwell`s equations and elasticity problems. 5. Implementation of algorithms: Development of a linear solver package (in C/C++). The main purpose of this project is to contribute in filling the gap between symmetric and positive definite (SPD) M-matrices and general SPD matrices, and, what is even more challenging, between general SPD matrices and non-symmetric and/or indefinite matrices. Besides the investigation of new classes of linear solvers it is also planned to develop a powerful tool kit that can be integrated in other research and commercial software packages as an essential part of the solver kernel.

The main goals of the FWF-Project No. P19170-N18 "Algebraic Multigrid for Vector-Field Problems" have been the improvement of existing and the design and the analysis of new Algebraic Multi-Grid (AMG) and multilevel preconditioning methods for solving large-scale systems of linear algebraic equations that stem from finite element discretization of (systems of) elliptic partial differential equations. The main focus has been on problems in solid and structural mechanics. In general, the presence of certain parameters makes the corresponding discrete problems extremely ill-conditined. As a consequence, highly efficient and robust preconditioning techniques are essential in order to be able to approximate the exact solution at the required level of accuracy and at a reasonable computational cost. This has become an important task when developing modern computer simulation software. In the following the project`s most important results (scientific advances) are summarized: The PhD student, Erwin Karer, who has been employed during the entire project runtime, has mainly been working on preconditioning techniques for linear elasticity problems. During the first half of the project runtime Mr. Karer and the PI improved and analyzed a method (originally proposed by the PI in 2007) for constructing the main components of AMG by exploiting low-rank matrices (so-called edge matrices). Based on this concept, an efficient preconditioner, also for anisotropic problems, has been developed. The obtained theoretical and numerical results have been published in the article "Algebraic multigrid for finite element elasticity equations: Determination of nodal dependence via edge matrices and two-level convergence" in the Int. J. Numer. Meth. Engng. (83(2010), pp. 642-670). During the second half of the project runtime, Mr. Karer, together with the PI, and with Prof. Ludmil Zikatanov (our collaboration partner from Penn State University, USA), considered elasticity problems describing the deformation of nearly incompressible solids. Here a main difficulty is to guarantee the stability and the optimal approximation properties of the finite element solution. In order to avoid so-called locking effects (that are observed when using standard discrtetization tools) we have been using a nonconforming method based on "reduced integration" as a starting point. We have devised a proper decomposition of the related finite element space. Further, we introduced a subspace correction method that yields a uniform (robust with respect to the Poisson ratio) preconditioner for this class of problems. An optimal order method for solving a crucial subproblem in this approach has been published by the PI (together with S. Tomar) in the article "Algebraic multilevel iteration method for lowest-order Raviart-Thomas space and applications" in the Int. J. Numer. Meth. Engng. (accepted). Further project-relevant publications of the PI, including a monography on "Robust Algebraic Multilevel Methods and Algorithms", published by Walter de Gruyter (2009), can be found in Section 4-"Attachments".