FWF Project P19170-N18
Runtime: 01.02.2007-31.01.2010

Project Team

Project Abstract

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.

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.

Keywords

  • Algebraic Multigrid
  • Multilevel Methods
  • Partial Differential Equations
  • Finite Element discretization
  • Preconditioning
  • Linear Solvers

Peer Reviewed Journal Publication

  • Karer, E.; Kraus, J. (2010) Algebraic multigrid for finite element elasticity equations: Determination of nodal dependence via edge matrices and two-level convergence. International Journal for Numerical Methods in Engineering.
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  • I. Georgiev, J. Kraus, S. Margenov (2008) Multilevel algorithms for Rannacher–Turek finite element approximation of 3D elliptic problems. Computing, Bd. 82, S. 217-239.
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  • J.K. Kraus, S.K. Tomar (2008) A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems. Numer. Linear Algebra Appl., Bd. 15, S. 417-438.
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  • J.K. Kraus, S.K. Tomar (2008) Multilevel preconditioning of two-dimensional elliptic problems discretized by a class of discontinuous Galerkin methods. SIAM J. Sci. Comput., Bd. 30 (2), S. 684-706.
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  • Kraus, J.K. (2008) Algebraic multigrid based on computational molecules, II: Linear elasticity problems. SIAM J. Sci. Comput., Bd. 30 (1), S. 505-524.
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  • I. Georgiev, J.K. Kraus, S. Margenov (2008) Multilevel preconditioning of rotated bilinear non-conforming FEM problems. Computers & Mathematics with Applications, Bd. 55, S. 2280-2294.
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  • J.K. Kraus, S.D. Margenov, J. Synka (2008) On the multilevel preconditioning ofCrouzeix-Raviart elliptic problems. Numer. Linear Algebra Appl., Bd. 15, S. 395-416.
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Stand-alone publications

  • E. Karer, Subspace Correction Methods for Linear Elasticity, PhD Thesis, Linz, November, 2011
  • J. Kraus and S. Margenov: Robust Algebraic Multilevel Methods and Algorithms. Radon Series Comp. Appl. Math., vol. 5, Walter de Gruyter, Berlin/NewYork, 2009.
  • J. Kraus: Algebraic multilevel methods for solving elliptic finite element equations with symmetric positive definite matrices. Habilitation thesis, Johannes Kepler University, 2008.
  • J. Kraus and S. Margenov: Multilevel methods for anisotropic elliptic problems. In Lectures on Advanced Computational Methods in Mechanics, Radon Series Comp. Appl. Math., vol. 1, J. Kraus and U. Langer eds., pp. 47-88, 2007.