Subspace Correction Methods for Indefinite Problems

Funding organizationAustrian Science Fund (FWF)
Funding programStand-alone Projects
Funding period36 months
Project numberP22989-N18
Starting DateDecember 1, 2010
Host organisationJohann Radon Institute for Computational and Applied Mathematics
Austrian Academy of Sciences
Principal investigatorJohannes Kraus

Description

Description

In this project we plan to develop and analyze new subspace correction (SC) methods for the numerical solution of coupled systems of partial differential equations (PDE). The focus is on nearly singular symmetric positive definite (SPD) and on indefinite problems.
We propose an integrated approach in which it is essential to use discretization techniques that preserve certain conservation laws, and to combine them with an adaptive solution process. In this way, one can design methods that perform optimally with respect to:

(i) accurate approximation of the unknown quantities;
(ii) obtaining the numerical solution in optimal time; and
(iii) scalability with respect to both, problem size and advances in computer hardware.

The present project has the following three interrelated Components (C1)–(C3) with a main emphasis on systems with highly oscillatory coefficients:

(C1):

SC methods for nearly incompressible elasticity and Stokes flow.

(C2):

SC methods for total variation minimization of discrete functionals arising in sparse data recovery.

(C3):

Auxiliary space and SC methods for elliptic problems with highly oscillatory coefficients.

The primary goal of the proposed research work is to contribute to extending the theory and applicability of subspace correction methods to the above-mentioned classes of problems.

Starting point of the research plan is the use and interplay of stable and accurate finite element schemes and of the efficient preconditioning of the related discrete problems.
In the present setting nonconforming and in particular discontinuous Galerkin (DG) finite element methods provide adequate discretization tools. Some of their most attractive properties and practical advantages over conforming methods are that

a) it is easy to extended DG methods to higher approximation order;
b) they are well suited to treat complex geometries in combination with unstructured and hybrid meshes;
c) they can be combined with any element type where the grids are also allowed to have hanging nodes;
d) they can easily handle adaptive strategies;
e) they have favorable properties in view of parallel computing.

A main disadvantage of DG discretizations is that they produce an excess of degrees of freedom (as compared to conforming methods of the same approximation order) which in general makes the solution of the arising linear systems more difficult and more time consuming. We therefore put strong efforts on devising new efficient and robust solution methods, covering wider classes of problems (see (C1)–(C3)) that arise from nonconforming and discontinuous Galerkin discretizations. The final aim is to adapt our methods to and to test them on industrial and multiphysics applications, e.g., in reservoir engineering, or in life science. Some of the problems in which we are particularly interested stem from micro-mechanics modeling of heterogeneous media, e.g., the modeling of fluid flow in porous media, the determination of the bio-mechanical properties of bones, or the reconstruction of (medical) images. Typically such problems involve parameters that lead to highly ill-conditioned systems of linear algebraic equations.

People

People

Members

  • Johannes Kraus (Principal Investigator)
  • Ivan Georgiev (PostDoc)
  • Nadir Bayramov (PhD Student)
  • Qingguo Hong (PostDoc)

Partners

  • R. Blaheta, Academy of Sciences of the Czech Republic
  • S. Margenov, Bulgarian Academy of Sciences
  • P. Vassilevski, Lawrence Livermore National Laboratory, USA
  • L. Zikatanov, The Pennsylvania State University, USA

Publications

Publications

PUBLICATIONS IN PEER-REVIEWED JOURNALS

  • B. Ayuso, I. Georgiev, J. Kraus, L. Zikatanov: A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations. Math. Model. Numer. Anal. (in press).
  • J. Brannick, Y. Chen, J. Kraus, L. Zikatanov: Algebraic multilevel preconditioners for the graph Laplacian based on matching in graphs. SIAM J. Numer. Anal. (in press).
  • J. Kraus, M. Lymbery, and S. Margenov: Robust multilevel methods for quadratic finite element anisotropic elliptic problems. Numer. Lin. Alg. Appl. (in press).
  • K. Gahalaut, J. Kraus, and S. Tomar: Multigrid methods for isogeometric discretization. Computer Methods in Applied Mechanics and Engineering, 253, 413-425, 2013.
  • J. Kraus, P. Vassilevski, and L. Zikatanov: Polynomial of best uniform approximation to 1/x and smoothing in two-level methods. Computational Methods in Applied Mathematics, Special issue devoted to Sergey Nepomnyaschikh, 12(4), 448-468, 2012.
  • P. Boyanova, I. Georgiev, S. Margenov, L. Zikatanov: Multilevel Preconditioning of Graph Laplacians: Polynomial Approximation of the Pivot Blocks Inverses, Mathematics and Computers in Simulation, 82, 1964-1971, 2012
  • J. Kraus: Additive Schur complement approximation and application to multilevel preconditioning. SIAM J. Sci. Comput. 34(6), A2872-A2895, 2012
  • J. Kraus and S. Tomar: Algebraic multilevel iteration method for lowest-order Raviart-Thomas space and applications. Int. J. Numer. Meth. Engng. 86, 1175-1196, 2011.

REFEREED PUBLICATIONS IN CONFERENCE PROCEEDINGS

  • E. Karer, J. Kraus, L. Zikatanov: A subspace correction method for nearly singular linear elasticity problems. In Proceedings of the 20th International Conference on Domain Decomposition Methods, UC San Diego, La Jolla, California, Lecture Notes in Computational Science and Engineering, Springer (to appear).
  • I. Georgiev, J. Kraus, Preconditioning of elasticity Problems with discontinuous Material Parameters, In Numerical Mathematics and Advanced Applications 2011, A. Cangiani et al. (eds.), Springer, 2013 (to appear)
  • J. Kraus, M. Lymbery, S. Margenov: Semi-coarsening AMLI preconditioning of higher order elliptic problems, Applications of Mathematics in Technical and Natural Sciences, M. Todorov edt., AIP Conference Proceedings, 1487, 30-41, 2012
  • J. Kraus: Additive Schur complement approximation for elliptic problems with oscillatory coefficients. In Large-Scale Scientific Computing, I. Lirkov, S. Margenov, and J. Wasniewski, eds., Lecture Notes in Computer Science, 7116, Springer, 52-59, 2012.
  • J. Kraus, M. Lymbery, S. Margenov: On the robustness of two-level preconditioners for quadratic FE orthotropic elliptic problems. In Large-Scale Scientific Computing, I. Lirkov, S. Margenov, and J. Wasniewski, eds., Lecture Notes in Computer Science, 7116, Springer, 582-589, 2012.
  • I. Georgiev, M. Lymbery, and S. Margenov, Analysis of the CBS constant for quadratic finite elements. In Numerical Methods and Applications, N. Kolkovska, I. Dimov, S. Dimova eds., Lecture Notes in Computer Science, 6046, Springer, 412-419, 2011.
  • B. Ayuso, I. Georgiev, J. Kraus, and L. Zikatanov: A simple preconditioner for the SIPG for discretizations of linear elasticity equations. In Numerical Methods and Applications, N. Kolkovska, I. Dimov, S. Dimova eds., Lecture Notes in Computer Science, 6046, Springer, 353-360, 2011.
  • I. Georgiev, J. Kraus, S. Margenov, Two-level Preconditioning for DG Discretizations of Scalar Elliptic Problems with Discontinuous Coefficients, Applications of Mathematics in Technical and Natural Sciences, AMiTaNS 2011, C. Christov, M. Todorov eds. AIP Conference Proceedings, 1404, 389-396, 2011.

Activities

SELECTED WORKSHOP AND CONFERENCE TALKS

CONFERENCE ORGANIZATION

  • Mini-symposium “Robust Multilevel and Multiscale Methods”, 6-th European Congress on Computational Methods in Applied Sciences and Engineering, Vienna, Austria, September 10-14, 2012.
    J. Kraus
  • Special session “Scalable Numerical Solution Methods for PDEs”, 21th International Conference on Domain Decomposition Methods (DD21), INRIA Rennes-Bretagne-Atlantique (France), June 25-29, 2012.
    J. Kraus
  • Special session “Robust Multilevel Methods and Multiscale Problems”, Fourth Conference of the Euro-American Consortium for Promoting the Application of Mathematics in Technical and Natural Sciences, St.St. Constantine and Helena, Varna, Bulgaria, June 11-16, 2012.
    I. Georgiev, J. Kraus
  • Workshop “Simulation of Flow in Porous Media and Applications in Waste Management and CO2 Sequestration” Linz, October 3-7, 2011
    J. Kraus