Stable Evaluation of Jacobian Matrices on Highly Refined Finite Element Meshes

In the adaptive finite element method, the solution of a p.d.e. is approximated by finer and finer meshes, which are controlled from error estimators. So, starting from a given coarse mesh, some elements are subdivided a couple of times. We investigate the question of avoiding instabilities which limit this process from the fact that nodal coordinates of one element coincide in more and more leading digits. To overcome this problem we demonstrate a simple mechanism for red subdivision of triangles (and hanging nodes) and a more sophisticated technique for general quadrilaterals.     

Solution of Large Scale Algebraic Matrix Riccati Equations by Use of Hierarchical Matrices

In previous papers, a class of hierarchical matrices (ℋ-matrices) is introduced which are data-sparse and allow an approximate matrix arithmetic of almost optimal complexity. Here, we investigate a new approach to exploit the ℋ-matrix structure for the solution of large scale Lyapunov and Riccati equations as they typically arise for optimal control problems where the constraint is a partial differential equation of elliptic type. This approach leads to an algorithm of linear-logarithmic complexity in the size of the matrices. Received July 30, 2002; revised December 16, 2002 Published online: April 22, 2003     

Hierarchical LU Decomposition-based Preconditioners for BEM

The adaptive cross approximation method can be used to efficiently approximate stiffness matrices arising from boundary element applications by hierarchical matrices. In this article an approximative LU decomposition in the same format is presented which can be used for preconditioning the resulting coefficient matrices efficiently. If the LU decomposition is computed with high precision, it may even be used as a direct yet efficient solver.     

Multilevel norms forH −1/2

We give some estimates related to finite element multilevel splittings and Sobolev norms of negative order. Basically, results for the positive order case are carried over by duality. In particular, semi-orthogonal splittings based on piecewise constants are studied for Sobolev spaces of order −1/2. Numerical experiments are provided for the screen problem.     

Computing Interpolation Weights in AMG Based on Multilevel Schur Complements

This paper presents a particular construction of neighborhood matrices to be used in the computation of the interpolation weights in AMG (algebraic multigrid). The method utilizes the existence of simple interpolation matrices (piecewise constant for example) on a hierarchy of coarse spaces (grids). Then one constructs by algebraic means graded away coarse spaces for any given fine-grid neighborhood. Next, the corresponding stiffness matrix is computed on this graded away mesh, and the actual neighborhood matrix is obtained by computing the multilevel Schur complement of this matrix where degrees of freedom outside the neighborhood have to be eliminated. The paper presents algorithmic details, provides model complexity analysis as well as some comparative tests of the quality of the resulting interpolation based on the multilevel Schur complements versus element interpolation based on the true element matrices.     

Applications of -Matrix Techniques in Micromagnetics

The variational model by Landau and Lifshitz is frequently used in the simulation of stationary micromagnetic phenomena. We consider the limit case of large and soft magnetic bodies, treating the associated Maxwell equation exactly via an integral operator . In numerical simulations of the resulting minimization problem, difficulties arise due to the imposed side-constraint and the unboundedness of the domain. We introduce a possible discretization by a penalization strategy. Here the computation of is numerically the most challenging issue, as it leads to densely populated matrices. We show how an efficient treatment of both and the corresponding bilinear form can be achieved by application of -matrix techniques.     

Algebraic Multigrid Based on Computational Molecules, 1: Scalar Elliptic Problems

We consider the problem of splitting a symmetric positive definite (SPD) stiffness matrix A arising from finite element discretization into a sum of edge matrices thereby assuming that A is given as the sum of symmetric positive semidefinite (SPSD) element matrices. We give necessary and sufficient conditions for the existence of an exact splitting into SPSD edge matrices and address the problem of best positive (nonnegative) approximation. Based on this disassembling process we present a new concept of ``strong' and ``weak' connections (edges), which provides a basis for selecting the coarse-grid nodes in algebraic multigrid methods. Furthermore, we examine the utilization of computational molecules (small collections of edge matrices) for deriving interpolation rules. The reproduction of edge matrices on coarse levels offers the opportunity to combine classical coarsening algorithms with effective (energy minimizing) interpolation principles yielding a flexible and robust new variant of AMG.     

Factorized Solution of Lyapunov Equations Based on Hierarchical Matrix Arithmetic

We investigate the numerical solution of large-scale Lyapunov equations with the sign function method. Replacing the usual matrix inversion, addition, and multiplication by formatted arithmetic for hierarchical matrices, we obtain an implementation that has linear-polylogarithmic complexity and memory requirements. The method is well suited for Lyapunov operators arising from FEM and BEM approximations to elliptic differential operators. With the sign function method it is possible to obtain a low-rank approximation to a full-rank factor of the solution directly. The task of computing such a factored solution arises, e.g., in model reduction based on balanced truncation. The basis of our method is a partitioned Newton iteration for computing the sign function of a suitable matrix, where one part of the iteration uses formatted arithmetic while the other part directly yields approximations to the full-rank factor of the solution. We discuss some variations of our method and its application to generalized Lyapunov equations. Numerical experiments show that the method can be applied to problems of order up to (10⁵) on workstations.     

Remarks on a posteriori error estimation for inaccurate finite element solutions

Usually, error estimators for adaptive refinement require exact discrete solutions. In this paper, we show how inaccurate solutions (e.g., iterative approximations) can be used, too. As a side remark we characterise iterative solution schemes that are particularly suited to producing good approximations for error estimators. This work was supported by Deutsche Forschungsgemeinschaft (Project Ha 1324/9).     

H 1-Galerkin Mixed Finite Element Method for the Regularized Long Wave Equation

In this paper, an H¹-Galerkin mixed finite element method is proposed for the 1-D regularized long wave (RLW) equation u_t+u_x+uu_x−δu_xxt=0. The existence of unique solutions of the semi-discrete and fully discrete H¹-Galerkin mixed finite element methods is proved, and optimal error estimates are established. Our method can simultaneously approximate the scalar unknown and the vector flux effectively, without requiring the LBB consistency condition. Finally, some numerical results are provided to illustrate the efficacy of our method.     

Numerical Integrations and Unit Resolution Multipliers for Domain Decomposition Methods with Nonmatching Grids

In this paper, we are concerned with the non-overlapping domain decomposition method (DDM) with nonmatching grids for three-dimensional problems. The weak continuity of the DDM solution on the interface is imposed by some Lagrange multiplier. We shall first analyze the influence of the numerical integrations over the interface on the (non-conforming) approximate solution. Then we will propose a simple approach to construct multiplier spaces, one of which can be simply spanned by some smooth basis functions with local compact supports, and thus makes the numerical integrations on the interface rather simple and inexpensive. Also it is shown this multiplier space can generate an optimal approximate solution. Numerical results are presented to compare the new method with the point to point method widely used in engineering.     

Reduction of Smith Normal Form Transformation Matrices

Smith normal form computations are important in group theory, module theory and number theory. We consider the transformation matrices for the Smith normal form over the integers and give a presentation of arbitrary transformation matrices for this normal form. Our main contribution is an algorithm that replaces already computed transformation matrices by others with small entries. We combine methods from lattice basis reduction with a procedure to reduce the sum of the squared entries of both transformation matrices. This algorithm performs well even for matrices of large dimensions.     

A Note on the Energy Norm for a Singularly Perturbed Model Problem

This paper considers a singularly perturbed reaction diffusion problem. It is investigated whether adaptive approaches are successful to design robust solution procedures. A key ingredient is the a posteriori error estimator. Since robust and mathematically analysed error estimation is possible in the energy norm, the focus is on this choice of norm and its implications. The numerical performance for several model problems confirms that the proposed adaptive algorithm (in conjunction with an energy norm error estimator) produces optimal results. Hence the energy norm is suitable for the purpose considered here. The investigations also provide valuable justification for forthcoming research. Received October 25, 2001; revised July 12, 2002 Published online: October 24, 2002     

An Alternative Algorithm for a Sliding Window ULV Decomposition

The ULV decomposition (ULVD) is an important member of a class of rank-revealing two-sided orthogonal decompositions used to approximate the singular value decomposition (SVD). The ULVD can be modified much faster than the SVD. When modifiying the ULVD, the accurate computation of the subspaces is required in certain time varying applications in signal processing. In this paper, we present an updating algorithm which is suitable for large scaled matrices of low rank and as effective as alternatives. The algorithm runs in O(n²) time.     

On the convergence of spectral multigrid methods for solving periodic problems

The spectral multigrid method combines the efficiencies of the spectral method and the multigrid method. In this paper, we show that various spectral multigrid methods have constant convergence rates (independent of the number of unknowns in the linear system, to be solved) in their multilevel iterations for solving periodic problems.     

Hierarchical Tensor-Product Approximation to the Inverse and Related Operators for High-Dimensional Elliptic Problems

The class of -matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the -matrix technique combined with the Kronecker tensor-product approximation (cf. [2, 20]) to represent the inverse of a discrete elliptic operator in a hypercube (0, 1)^dd in the case of a high spatial dimension d. In this data-sparse format, we also represent the operator exponential, the fractional power of an elliptic operator as well as the solution operator of the matrix Lyapunov-Sylvester equation. The complexity of our approximations can be estimated by (dn log ^qn), where N=n^d is the discrete problem size.     

Tensor Product Multigrid for Maxwell's Equation with Aligned Anisotropy

When simulating electromagnetic phenomena in symmetric cavities, it is often possible to exploit the symmetry in order to reduce the dimension of the problem, thereby reducing the amount of work necessary for its numerical solution. Usually, this reduction leads not only to a much lower number of unknowns in the discretized system, but also changes the behaviour of the coefficients of the differential operator in an unfavourable way, usually leading to the transformed system being not elliptic with respect to norms corresponding to two-dimensional space, thus limiting the use of standard multigrid techniques. In this paper, we introduce a robust multigrid method for Maxwell's equation in two dimensions that is especially suited for coefficients resulting from coordinate transformations, i.e. that are aligned with the coordinate axes. Using a variant of the technique introduced in [5], we can prove robustness of the multigrid method for domains of tensor-product structure and coefficients depending on only one of the coordinates. Received July 17, 2000; revised October 27, 2000     

Fractal-like Matrices

We introduce special sparse n×n-matrices (n=2 ^k ) containing up to 3 ^k ≈n ^1.585 nonzero elements. Their structure is inherited from the famous Sierpinski triangle and is not sensitive to matrix multiplication and inversion. The arithmetical complexity of taking product or inverse of such matrices is proved to be O(n ²).     

The hierarchial preconditioning on unstructured grids

We present the implementation of two hierarchically preconditioned methods for the fast solution of mesh equations that approximate 2D-elliptic boundary value problems on unstructured quasi uniform triangulations. Based on the fictitious space approach the original problem can be embedded into an auxiliary one, where both the hierarchical grid information and the preconditioner are well defined. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as thebpx-preconditioned cg-iteration having optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods which can be of importance in industrial engineering, where often only the nodal coordinates and the element connectivity of the underlying (fine) discretization are available.     

On the reference mapping for quadrilateral and hexahedral finite elements on multilevel adaptive grids

We study the properties of the reference mapping for quadrilateral and hexahedral finite elements. We consider multilevel adaptive grids with possibly hanging nodes which are typically generated by adaptive refinement starting from a regular coarse grid. It turns out that for such grids the reference mapping behaves – up to a perturbation depending on the mesh size – like an affine mapping. As an application, we prove optimal estimates of the interpolation error for discontinuous mapped -elements on quadrilateral and hexahedral grids.