Accelerated monotone iterative methods for finite difference equations of reaction-diffusion

This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate the various rates of convergence of the iterations. Received November 2, 1995 / Revised version received February 10, 1997     

Adaptive finite element methods for elliptic equations with non-smooth coefficients

Summary. We consider a second-order elliptic equation with discontinuous or anisotropic coefficients in a bounded two- or three dimensional domain, and its finite-element discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients. Received February 5, 1999 / Published online March 16, 2000     

Asymptotic stability analysis of Runge-Kutta methods for nonlinear systems of delay differential equations

Summary. We consider systems of delay differential equations (DDEs) of the form with the initial condition . Recently, Torelli [10] introduced a concept of stability for numerical methods applied to dissipative nonlinear systems of DDEs (in some inner product norm), namely RN-stability, which is the straighforward generalization of the wellknown concept of BN-stability of numerical methods with respect to dissipative systems of ODEs. Dissipativity means that the solutions and corresponding to different initial functions and , respectively, satisfy the inequality , and is guaranteed by suitable conditions on the Lipschitz constants of the right-hand side function . A numerical method is said to be RN-stable if it preserves this contractivity property. After showing that, under slightly more stringent hypotheses on the Lipschitz constants and on the delay function , the solutions and are such that , in this paper we prove that RN-stable continuous Runge-Kutta methods preserve also this asymptotic stability property. Received March 29, 1996 / Revised version received August 12, 1996     

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Summary. In previous works [21–23] we proposed the use of [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis. Received December 19, 1995 / Revised version received September 15, 1997     

Qualocation methods for elliptic boundary integral equations

Summary. The qualocation methods developed in this paper, with spline trial and test spaces, are suitable for classes of boundary integral equations with convolutional principal part, on smooth closed curves in the plane. Some of the methods are suitable for all strongly elliptic equations; that is, for equations in which the even symbol part of the operator dominates. Other methods are suitable when the odd part dominates. Received December 27, 1996 / Revised version received April 14, 1997     

Spline qualocation methods for variable-coefficient elliptic equations on curves

Summary. Here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type. The analysis exploits localization in the form of frozen coefficients, pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators, a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders. Numerical examples show the predicted high orders of convergence. Received January 29, 1998 / Published online: June 29, 1999     

On hybrid iterative methods for nonsymmetric systems of linear equations

Summary. Hybrid methods for the solution of systems of linear equations consist of a first phase where some information about the associated coefficient matrix is acquired, and a second phase in which a polynomial iteration designed with respect to this information is used. Most of the hybrid algorithms proposed recently for the solution of nonsymmetric systems rely on the direct use of eigenvalue estimates constructed by the Arnoldi process in Phase I. We will show the limitations of this approach and propose an alternative, also based on the Arnoldi process, which approximates the field of values of the coefficient matrix and of its inverse in the Krylov subspace. We also report on numerical experiments comparing the resulting new method with other hybrid algorithms. Received May 27, 1993 / Revised version received November 14, 1994     

Minimal decompositions and iterative methods

Summary. The aim of this paper is to prove some Babuška–Brezzi type conditions which are involved in the mortar spectral element discretization of the Stokes problem, for several cases of nonconforming domain decompositions. ID=" <E5>Dedicated to Olof B. Widlund on the occasion of his 60th birthday</E5>     

Robust iterative methods for elliptic problems with highly varying coefficients in thin substructures

Summary. In this paper we introduce a class of robust multilevel interface solvers for two-dimensional finite element discrete elliptic problems with highly varying coefficients corresponding to geometric decompositions by a tensor product of strongly non-uniform meshes. The global iterations convergence rate is shown to be of the order with respect to the number of degrees of freedom on the single subdomain boundaries, uniformly upon the coarse and fine mesh sizes, jumps in the coefficients and aspect ratios of substructures. As the first approach, we adapt the frequency filtering techniques [28] to construct robust smoothers on the highly non-uniform coarse grid. As an alternative, a multilevel averaging procedure for successive coarse grid correction is proposed and analyzed. The resultant multilevel coarse grid preconditioner is shown to have (in a two level case) the condition number independent of the coarse mesh grading and jumps in the coefficients related to the coarsest refinement level. The proposed technique exhibited high serial and parallel performance in the skin diffusion processes modelling [20] where the high dimensional coarse mesh problem inherits a strong geometrical and coefficients anisotropy. The approach may be also applied to magnetostatics problems as well as in some composite materials simulation. Received December 27, 1994     

Overlapping Schwarz methods for Maxwell''s equations in three dimensions

Summary. A two-level overlapping Schwarz method is considered for a Nédélec finite element approximation of 3D Maxwell's equations. For a fixed relative overlap, the condition number of the method is bounded, independently of the mesh size of the triangulation and the number of subregions. Our results are obtained with the assumption that the coarse triangulation is quasi-uniform and, for the Dirichlet problem, that the domain is convex. Our work generalizes well–known results for conforming finite elements for second order elliptic scalar equations. Numerical results for one and two-level algorithms are also presented. Received November 11, 1997 / Revised version received May 26, 1999 / Published online June 21, 2000     

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

Summary. We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers. Received August 5, 2000 / Published online June 20, 2001     

A multilevel algorithm for the solution of second order elliptic differential equations on sparse grids

A multilevel algorithm is presented that solves general second order elliptic partial differential equations on adaptive sparse grids. The multilevel algorithm consists of several V-cycles in - and -direction. A suitable discretization provide that the discrete equation system can be solved in an efficient way. Numerical experiments show a convergence rate of order for the multilevel algorithm. Received April 19, 1996 / Revised version received December 9, 1996     

Regular solutions of nonlinear differential-algebraic equations and their numerical determination

For a general class of nonlinear (possibly higher index) differential-algebraic equations we show existence and uniqueness of solutions. These solutions are regular in the sense that Newton's method will converge locally and quadratically. On the basis of the presented theoretical results, numerical methods for the determination of consistent initial values and for the computation of regular solutions are developed. Several numerical examples are included. Received July 4, 1996 / Revised version received May 12, 1997     

Boundary element methods for Maxwell's equations on non-smooth domains

Summary. Variational boundary integral equations for Maxwell's equations on Lipschitz surfaces in are derived and their well-posedness in the appropriate trace spaces is established. An equivalent, stable mixed reformulation of the system of integral equations is obtained which admits discretization by Galerkin boundary elements based on standard spaces. On polyhedral surfaces, quasioptimal asymptotic convergence of these Galerkin boundary element methods is proved. A sharp regularity result for the surface multipliers on polyhedral boundaries with plane faces is established. Received January 5, 2001 / Revised version received August 6, 2001 / Published online December 18, 2001 Correspondence to: C. Schwab     

Instability of Runge-Kutta methods when applied to linear systems of delay differential equations

Summary. This paper investigates the stability of Runge-Kutta methods when they are applied to the complex linear system of delay differential equations , where . We prove that no Runge-Kutta method preserves asymptotic stability. Received January 24, 2000 / Revised version received July 19, 2000 / Published online June 7, 2001     

Least-squares mixed finite element methods for non-selfadjoint elliptic problems: I. Error estimates

Summary. A least-squares mixed finite element method for general second-order non-selfadjoint elliptic problems in two- and three-dimensional domains is formulated and analyzed. The finite element spaces for the primary solution approximation and the flux approximation consist of piecewise polynomials of degree and respectively. The method is mildly nonconforming on the boundary. The cases and are studied. It is proved that the method is not subject to the LBB-condition. Optimal - and -error estimates are derived for regular finite element partitions. Numerical experiments, confirming the theoretical rates of convergence, are presented. Received October 15, 1993 / Revised version received August 2, 1994     

Parallel algorithms for maximal monotone operators of local type

Summary. This paper presents general algorithms for the parallel solution of finite element problems associated with maximal monotone operators of local type. The latter concept, which is also introduced here, is well suited to capture the idea that the given operator is the discretization of a differential operator that may involve nonlinearities and/or constraints as long as those are of a local nature. Our algorithms are obtained as a combination of known algorithms for possibly multi-valued maximal monotone operators with appropriate decompositions of the domain. This work extends a method due to two of the authors in the single-valued and linear case. Received April 25, 1994     

Block and asynchronous two-stage methods for mildly nonlinear systems

Block parallel iterative methods for the solution of mildly nonlinear systems of equations of the form are studied. Two-stage methods, where the solution of each block is approximated by an inner iteration, are treated. Both synchronous and asynchronous versions are analyzed, and both pointwise and blockwise convergence theorems provided. The case where there are overlapping blocks is also considered. The analysis of the asynchronous method when applied to linear systems includes cases not treated before in the literature. Received June 5, 1997 / Revised version received December 29, 1997     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods

Summary. Given a nonsingular matrix , and a matrix of the same order, under certain very mild conditions, there is a unique splitting , such that . Moreover, all properties of the splitting are derived directly from the iteration matrix . These results do not hold when the matrix is singular. In this case, given a matrix and a splitting such that , there are infinitely many other splittings corresponding to the same matrices and , and different splittings can have different properties. For instance, when is nonnegative, some of these splittings can be regular splittings, while others can be only weak splittings. Analogous results hold in the symmetric positive semidefinite case. Given a singular matrix , not for all iteration matrices there is a splitting corresponding to them. Necessary and sufficient conditions for the existence of such splittings are examined. As an illustration of the theory developed, the convergence of certain alternating iterations is analyzed. Different cases where the matrix is monotone, singular, and positive (semi)definite are studied. Received September 5, 1995 / Revised version received April 3, 1996