On the convergence of the Bermúdez-Moreno algorithm with constant parameters

Summary. Bermúdez-Moreno [5] presents a duality numerical algorithm for solving variational inequalities of the second kind. The performance of this algorithm strongly depends on the choice of two constant parameters. Assuming a further hypothesis of the inf-sup type, we present here a convergence theorem that improves on the one presented in [5]: we prove that the convergence is linear, and we give the expression of the asymptotic error constant and the explicit form of the optimal parameters, as a function of some constants related to the variational inequality. Finally, we present some numerical examples that confirm the theoretical results. Received June 28, 1999 / Revised version received February 19, 2001 / Published online October 17, 2001     

Stability of the 8-tetrahedra shortest-interior-edge partitioning method

We consider a tetrahedron partitioning method, known in the literature as the 8-tetrahedra shortest-interior-edge partition. For this method, which is a variant of Freudenthal’s algorithm in three space dimensions, we prove that the infinite series of refined meshes (for any given initial mesh) is stable in the sense that the degree of degeneracy of the cells remains bounded. We give an explicit estimate in terms of a standard shape quality measure introduced by Liu and Joe. Furthermore, we show that our estimate is sharp. The estimate also holds for Freudenthal’s algorithm (in three space dimensions) provided that it is initialized appropriately. Numerical experiments confirm our result as well as its sharpness.     

On computation of solution curves for semilinear elliptic problems

We consider computation of solution curves for semilinear elliptic equations. In case solution is stable, we present an algorithm with monotone convergence, which is a considerable improvement of the corresponding schemes in [4] and [5]. For the unstable solutions, we show how to construct a fourth-order evolution equation, for which the same solution will be stable.     

A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem

In this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf-sup constant is available, which is confirmed by some numerical results.     

Superconvergence and a posteriori error estimation for triangular mixed finite elements

Summary. In this paper,we prove superconvergence results for the vector variable when lowest order triangular mixed finite elements of Raviart-Thomas type [17] on uniform triangulations are used, i.e., that the -distance between the approximate solution and a suitable projection of the real solution is of higher order than the -error. We prove results for both Dirichlet and Neumann boundary conditions. Recently, Duran [9] proved similar results for rectangular mixed finite elements, and superconvergence along the Gauss-lines for rectangular mixed finite elements was considered by Douglas, Ewing, Lazarov and Wang in [11], [8] and [18]. The triangular case however needs some extra effort. Using the superconvergence results, a simple postprocessing of the approximate solution will give an asymptotically exact a posteriori error estimator for the -error in the approximation of the vector variable. Received December 6, 1992 / Revised version received October 2, 1993     

An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems

In this paper, we modify the adaptive wavelet algorithm from Gantumur et al. [An optimal adaptive wavelet method without coarsening of the iterands, Technical Report 1325, Department of Mathematics, Utrecht University, March 2005, Math. Comp., to appear] so that it applies directly, i.e., without forming the normal equation, not only to self-adjoint elliptic operators but also to operators of the form L=A+B$L=A+B$, where A is self-adjoint elliptic and B is compact, assuming that the resulting operator equation is well posed. We show that the algorithm has optimal computational complexity.     

Transpose-free formulations of Lanczos-type methods for nonsymmetric linear systems

We present a transpose-free version of the nonsymmetric scaled Lanczos procedure. It generates the same tridiagonal matrix as the classical algorithm, using two matrix–vector products per iteration without accessing A^T. We apply this algorithm to obtain a transpose-free version of the Quasi-minimal residual method of Freund and Nachtigal [15] (without look-ahead), which requires three matrix–vector products per iteration. We also present a related transpose-free version of the bi-conjugate gradients algorithm. This revised version was published online in June 2006 with corrections to the Cover Date.     

Pivoting strategy for rank-one modification ofLDM t-like factorization

The rank-one modification algorithm of theLDM ^t factorization was given by Bennett [1]. His method, however, could break down even when the matrix is nonsingular and well-conditioned. We introduce a pivoting strategy for avoiding possible break-down as well as for suppressing error growth in the modification process. The method is based on a symbolic formula of the rank-one modification of the factorization of a possibly singular nonsymmetric matrix. A new symbolic formula is also obtained for the inverses of the factor matrices. Repeated application of our method produces theLDM ^t-like product form factorization of a matrix. A numerical example is given to illustrate our pivoting method. An incomplete factorization algorithm is also introduced for updating positive definite matrix useful in quasi-Newton methods, in which the Fletcher and Powell algorithm [2] and the Gill, Murray and Saunders algorithm [4] are usually used.This paper is presented at the Japan SIAM Annual Meeting held at University of Tokyo, Japan, October 7–9, 1991.     

Restriction matrices for numerically exploiting symmetry

In this paper we develop a technique for exploiting symmetry in the numerical treatment of boundary value problems (BVP) and eigenvalue problems which are invariant under a finite group of congruences of . This technique will be based upon suitable restriction matrices strictly related to a system of irreducible matrix representation of . Both Abelian and non-Abelian finite groups are considered. In the framework of symmetric Galerkin boundary element method (SGBEM), where the discretization matrices are typically full, to increase the computational gain we couple Panel Clustering Method [30] and Adaptive Cross Approximation algorithm [13] with restriction matrices introduced in this paper, showing some numerical examples. Applications of restriction matrices to SGBEM under the weaker assumption of partial geometrical symmetry, where the boundary has disconnected components, one of which is invariant, are proposed. The paper concludes with several numerical tests to demonstrate the effectiveness of the introduced technique in the numerical resolution of Dirichlet or Neumann invariant BVPs, in their differential or integral formulation.      

An asymptotically optimal Schur complement reduction for the Stokes equation

Summary. In this paper we develop an efficient Schur complement method for solving the 2D Stokes equation. As a basic algorithm, we apply a decomposition approach with respect to the trace of the pressure. The alternative stream function-vorticity reduction is also discussed. The original problem is reduced to solving the equivalent boundary (interface) equation with symmetric and positive definite operator in the appropriate trace space. We apply a mixed finite element approximation to the interface operator by iso triangular elements and prove the optimal error estimates in the presence of stabilizing bubble functions. The norm equivalences for the corresponding discrete operators are established. Then we propose an asymptotically optimal compression technique for the related stiffness matrix (in the absence of bubble functions) providing a sparse factorized approximation to the Schur complement. In this case, the algorithm is shown to have an optimal complexity of the order , q = 2 or q = 3, depending on the geometry, where N is the number of degrees of freedom on the interface. In the presence of bubble functions, our method has the complexity arithmetical operations. The Schur complement interface equation is resolved by the PCG iterations with an optimal preconditioner. Received March 20, 1996 / Revised version received October 28, 1997     

On the implementation of boundary conditions for the method of lines

The method of lines for difference approximations of hyperbolic first order systems of partial differential equations is analyzed. The approximations are based on strictly semibounded difference operators including high order ones. The formulation of the ODE-system requires that the implementation of the boundary conditions is done carefully. We shall illustrate how different ways of implementation give rise to different stability properties. In particular, we derive a way of implementation that leads to an approximation that is strongly stable. It has been an open problem, whether for semidiscrete approximations with this strong stability property, the timestep for the ODE-solver is governed by the Cauchy problem. We present a counterexample showing that it is not. The analysis presented in this paper also serves as an illustration of the significant difference between different stability concepts.     

An Arnoldi-Extrapolation algorithm for computing PageRank

The Arnoldi-type algorithm proposed by Golub and Greif [G. Golub, C. Greif, An Arnoldi-type algorithm for computing PageRank, BIT 46 (2006) 759-771] is a restarted Krylov subspace method for computing PageRank. However, this algorithm may not be efficient when the damping factor is high and the dimension of the search subspace is small. In this paper, we first develop an extrapolation method based on Ritz values. We then consider how to periodically knit this extrapolation method together with the Arnoldi-type algorithm. The resulting algorithm is the Arnoldi-Extrapolation algorithm. The convergence of the new algorithm is analyzed. Numerical experiments demonstrate the numerical behavior of this algorithm.     

Error analysis of an enhanced DtN-FE method for exterior scattering problems

In this work we analyze the convergence of the high-order Enhanced DtN-FEM algorithm, described in our previous work (Nicholls and Nigam, J. Comput. Phys. 194:278–303, 2004), for solving exterior acoustic scattering problems in . This algorithm consists of using an exact Dirichlet-to-Neumann (DtN) map on a hypersurface enclosing the scatterer, where the hypersurface is a perturbation of a circle, and, in practice, the perturbation can be very large. Our theoretical work had shown the DtN map was analytic as a function of this perturbation. In the present work, we carefully analyze the error introduced by virtue of using this algorithm. Specifically, we give a full account of the error introduced by truncating the DtN map at a finite order in the perturbation expansion, and study the well-posedness of the associated formulation. During computation, the Fourier series of the Dirichlet data on the artificial boundary must be truncated. To deal with the ensuing loss of uniqueness of solutions, we propose a modified DtN map, and prove well-posedness of the resulting problem. We quantify the spectral error introduced due to this truncation of the data. The key tools in the analysis include a new theorem on the analyticity of the DtN map in a suitable Sobolev space, and another on the perturbation of non-self-adjoint Fredholm operators.     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

Trigonometric Hermite wavelet approximation for the integral equations of second kind with weakly singular kernel

This paper is concerned with a trigonometric Hermite wavelet Galerkin method for the Fredholm integral equations with weakly singular kernel. The kernel function of this integral equation considered here includes two parts, a weakly singular kernel part and a smooth kernel part. The approximation estimates for the weakly singular kernel function and the smooth part based on the trigonometric Hermite wavelet constructed by E. Quak [Trigonometric wavelets for Hermite interpolation, Math. Comp. 65 (1996) 683–722] are developed. The use of trigonometric Hermite interpolant wavelets for the discretization leads to a circulant block diagonal symmetrical system matrix. It is shown that we only need to compute and store O(N)$\mathrm{O}(N)$ entries for the weakly singular kernel representation matrix with dimensions N²$N^2$ which can reduce the whole computational cost and storage expense. The computational schemes of the resulting matrix elements are provided for the weakly singular kernel function. Furthermore, the convergence analysis is developed for the trigonometric wavelet method in this paper.     

On the convergence of multi-grid methods with transforming smoothers

Summary In the present paper we give a convergence theory for multi-grid methods with transforming smoothers as introduced in [31] applied to a general system of partial differential equations. The theory follows Hackbusch's approach for scalar pde and allows a convergence proof for some well-known multi-grid methods for Stokes- and Navier-Stokes equations as DGS by Brandt-Dinar, [5], TILU from [31] and the SIMPLE-methods by Patankar-Spalding, [23].This work was supported in part by Deutsche Forschungsgemeinschaft     

Preconditioning cubic spline collocation discretizations of elliptic equations

Summary. This work considers the uniformly elliptic operator defined by in (the unit square) with boundary conditions: on and on and its discretization based on Hermite cubic spline spaces and collocation at the Gauss points. Using an interpolatory basis with support on the Gauss points one obtains the matrix . We discuss the condition numbers and the distribution of -singular values of the preconditioned matrices where is the stiffness matrix associated with the finite element discretization of the positive definite uniformly elliptic operator given by in with boundary conditions: on on . The finite element space is either the space of continuous functions which are bilinear on the rectangles determined by Gauss points or the space of continuous functions which are linear on the triangles of the triangulation of using the Gauss points. When we obtain results on the eigenvalues of . In the general case we obtain bounds and clustering results on the -singular values of . These results are related to the results of Manteuffel and Parter [MP], Parter and Wong [PW], and Wong [W] for finite element discretizations as well as the results of Parter and Rothman [PR] for discretizations based on Legendre Spectral Collocation. Received January 1, 1994 / Revised version received February 7, 1995     

Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems

Summary. In this paper, we consider some nonlinear inexact Uzawa methods for iteratively solving linear saddle-point problems. By means of a new technique, we first give an essential improvement on the convergence results of Bramble-Paschiak-Vassilev for a known nonlinear inexact Uzawa algorithm. Then we propose two new algorithms, which can be viewed as a combination of the known nonlinear inexact Uzawa method with the classical steepest descent method and conjugate gradient method respectively. The two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved, including the preconditioned Schur complement. Numerical results of the algorithms applied for the Stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms. Received December 8, 1999 / Revised version received September 8, 2001 / Published online March 8, 2002 RID="*" ID="*" The work of this author was partially supported by a grant from The Institute of Mathematical Sciences, CUHK RID="**" ID="**" The work of this author was partially supported by Hong Kong RGC Grants CUHK 4292/00P and CUHK 4244/01P     

On truncated incomplete decompositions

In the present paper we introduce truncated incomplete decompositions (TrILU) for constant coefficient matrices. This new ILU variant saves most of the memory and work usually needed to compute and store the factorization. Further it improves the smoothing and preconditioning properties of standard ILU-decompositions. Besides describing the algorithm, we give theoretical results concerning stability and convergence as well as the smoothing property and robustness for TrILU smoothing in a multi-grid method. Further, we add numerical results of TrILU as smoother in a multi-grid method and as preconditioner in a pcg-method fully confirming the theoretical results.This work was supported by Deutsche Forschungsgemeinschaft.     

A method for approximation of the exponential map in semidirect product of matrix Lie groups and some applications

In this paper we explore the computation of the matrix exponential in a manner that is consistent with Lie group structure. Our point of departure is the decomposition of Lie algebra as the semidirect product of two Lie subspaces and an application of the Baker-Campbell-Hausdorff formula. Our results extend the results in Iserles and Zanna (2005) [2], Zanna and Munthe-Kaas(2001/02) [4] to a range of Lie groups: the Lie group of all solid motions in Euclidean space, the Lorentz Lie group of all solid motions in Minkowski space and the group of all invertible (upper) triangular matrices. In our method, the matrix exponential group can be computed by a less computational cost and is more accurate than the current methods. In addition, by this method the approximated matrix exponential belongs to the corresponding Lie group.