Maria Lymbery

Preconditioning Heterogeneous

In this paper we propose and analyze a preconditioner for a system arising from a mixed finite element approximation of second-order elliptic problems describing processes in highly heterogeneous media. Our approach uses the technique of multilevel methods (see, e.g., [P. Vassilevski, Multilevel Block Factorization Preconditioners: Matrix-Based Analysis and Algorithms for Solving Finite Element Equations, Springer, New York, 2008]) and the recently proposed preconditioner based on additive Schur complement approximation by J. Kraus [SIAM J. Sci. Comput., 34 (2012), pp. A2872--A2895]. The main results are the design, study, and numerical justification of iterative algorithms for these problems that are robust with respect to the contrast of the media, defined as the ratio between the maximum and minimum values of the coefficient of the problem. Numerical tests provide experimental evidence for the high quality of the preconditioner and its desired robustness with respect to the material contrast. Such resu...

\boldsymbol{H}(\mathrm{div})

In this paper the idea of auxiliary space multigrid (ASMG) methods is introduced. The construction is based on a two-level block factorization of local (finite element stiffness) matrices associated with a partitioning of the domain into overlapping or non-overlapping subdomains. The two-level method utilizes a coarse-grid operator obtained from additive Schur complement approximation (ASCA). Its analysis is carried out in the framework of auxiliary space preconditioning and condition number estimates for both, the two-level preconditioner, as well as for the ASCA are derived. The two-level method is recursively extended to define the ASMG algorithm. In particular, so-called Krylov-cycles are considered. The theoretical results are supported by a representative collection of numerical tests which further demonstrate the efficiency of the new algorithm for multiscale problems.

Problems by Additive Schur Complement Approximation and Applications

We study the behavior of the CBS constant as a quality measure for hierarchical two-level splittings of quadratic FEM stiffness matrices. The article is written in the spirit of [3] where the focus is on the robustness with respect to mesh and coefficient anisotropy. The considered splittings are: Differences and Aggregates (DA); First Reduce (FR); and hierarchical P-decomposition (P). The presented results show sufficient conditions for the existence of optimal order Algebraic Multi-Level Iteration (AMLI) preconditioners.

Auxiliary space multigrid method based on additive Schur complement approximation

AstractWhile a large amount of papers are dealing with robust multilevel methods and algorithms for linear FEM elliptic systems, the related higher order FEM problems are much less studied. Moreover, we know that the standard hierarchical basis two-level splittings deteriorate for strongly anisotropic problems. A first robust multilevel preconditioner for higher order FEM systems obtained after discretizations of elliptic problems with an anisotropic diffusion tensor is presented in this paper. We study the behavior of the constant in the strengthened CBS inequality for semi-coarsening mesh refinement which is a quality measure for hierarchical two-level splittings of the considered biquadratic FEM stiffness matrices. The presented new theoretical estimates are confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). In the paper we consider also the problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level splittings. Combining the proven uniform estimates with the theory of the Algebraic MultiLevel Iteration (AMLI) methods we obtain an optimal order multilevel algorithm whose total computational cost is proportional to the size of the discrete problem with a proportionality constant independent of the anisotropy ratio.

Analysis of the CBS constant for quadratic finite elements

We study the construction of subspaces for quadratic FEM orthotropic elliptic problems with a focus on the robustness with respect to mesh and coefficient anisotropy. In the general setting of an arbitrary elliptic operator it is known that standard hierarchical basis (HB) techniques do not result in splittings in which the angle between the coarse space and its (hierarchical) complement is uniformly bounded with respect to the ratio of anisotropy. In this paper we present a robust splitting of the finite element space of continuous piecewise quadratic functions for the orthotropic problem. As a consequence of this result we obtain also a uniform condition number bound for a special sparse Schur complement approximation. Further we construct a uniform preconditioner for the pivot block with optimal order of computational complexity.

Robust semi-coarsening multilevel preconditioning of biquadratic FEM systems

This paper discusses a class of multilevel preconditioner s based on approximate block factorization for conforming finite element methods (FEM) empl oying quadratic trial and test functions. The main focus is on di ffusion problems governed by a scalar elliptic partial di fferential equation (PDE) with a strongly anisotropic coe fficient tensor. The proposed method provides a high robustness with respect to non-grid-aligned anisotropy, which is achieved by the interaction of the following components: (i) an additive Schur complement approximatio n to construct the coarse-grid operator; (ii) a global block (Jacobi or Gauss-Seidel) smoother compl e enting the coarse-grid correction based on (i); and (iii) utilization of an augmented coarse grid, wh ich enhances the e fficiency of the interplay between (i) and (ii); The performed analysis indicates the h igh robustness of the resulting two-level method. Moreover, numerical tests with a nonlinear algebra ic multilevel iteration (AMLI) method demonstrate that the presented two-level method can be appl ied successfully in the recursive construction of uniform multilevel preconditioners of optima l or nearly optimal order of computational complexity.

On the robustness of two-level preconditioners for quadratic FE orthotropic elliptic problems

The present paper presents the construction of a robust multilevel preconditioner for anisotropic bicubic finite element (FE) elliptic problems. More precisely, the behavior of the constant in the strengthened CBS inequality, which is important for studying (approximate) block factorizations of FE stiffness matrices, is analyzed in the case when the underlying conforming FE space consists of piecewise bicubic functions, and is decomposed according to hierarchical splittings that are based on semi-coarsening of the FE mesh. The presented theoretical estimates are further confirmed by numerically computed CBS constants for a rich set of parameters (coarsening factor and anisotropy ratio). The problem of solving efficiently systems with the pivot block matrices arising in the hierarchical basis two-level matrices is also addressed in this paper. Finally, combining the proven uniform estimates with the theory of the Algebraic Multilevel Iteration (AMLI) methods an optimal order multilevel algorithm whose tota...

Robust multilevel methods for quadratic finite element anisotropic elliptic problems

In the present study we demonstrate the construction of a robust multilevel preconditioner for biquadratic FE elliptic problems. In the general setting of an arbitrary elliptic operator it is well known that the standard hierarchical basis two‐level splittings for higher order FEM elliptic systems deteriorate with increasing the anisotropy ratio. An alternative approach resulting in a robust hierarchical two‐level splitting of the finite element space of continuos piecewise biquadratic functions involves the semi‐coarsening mesh procedure. This evokes us to analyze the behavior of the constant in the strengthened CBS inequality, which is a quality measure for hierarchical two‐level splittings of the FEM stiffness matrices, for the particular case of balanced semi‐coarsening mesh refinement. We present new theoretical estimates which further we support by numerically computed CBS constants over a rich set of parameters (coarsening factor and anisotropy ratio). An optimal order multilevel algorithm is const...

Semi-coarsening AMLI preconditioning of higher order elliptic problems

Abstract This study proposes a new preconditioning strategy for symmetric positive (semi-)definite SP(S)D matrices referred to as incomplete factorization by local exact factorization (ILUE). The investigated technique is based on exact LU decomposition of small-sized local matrices associated with a splitting of the domain into overlapping or non-overlapping subdomains. The ILUE preconditioner is defined and its relative condition number estimated. Numerical tests on linear systems arising from the finite element (FE) discretization of a second order elliptic boundary value problem in mixed form demonstrate the advantage of the new algorithm, even for problems with highly oscillatory permeability coefficients, against the classical ILU( p ) and ILUT( τ ) incomplete factorization preconditioners.

Robust Balanced Semi‐coarsening AMLI Preconditioning of Biquadratic FEM Systems

The robust preconditioning of linear systems of algebraic equations arising from discretizations of partial differential equations (PDE) is a fastly developing area of scientific research. In many applications these systems are very large, sparse and therefore it is vital to construct (quasi-)optimal iterative methods that converge independently of problem parameters.