Multi-level meshless methods based on direct multi-elliptic interpolation

A short overview on the direct multi-elliptic interpolation and the related meshless methods for solving partial differential equations is given. A new technique is proposed which produces a biharmonic interpolation along the boundary and solves the original problem inside the domain. An error estimation is also derived. To implement the method, quadtree-based multi-level methods are used. The approach avoids the use of large, dense and ill-conditioned matrices and significantly reduces the computational cost.     

A new fitted operator finite difference method to solve systems of evolutionary reaction-diffusion equations

Abstract

In recent years, fitted operator finite difference methods (FOFDMs) have been developed for numerous types of singularly perturbed ordinary differential equations. The construction of most of these methods differed though the final outcome remained similar. The most crucial aspect was how the difference operator was designed to approximate the differential operator in question. Very often the approaches for constructing these operators had limited scope in the sense that it was difficult to extend them to solve even simple one-dimensional singularly perturbed partial differential equations. However, in some of our most recent work, we have successfully designed a class of FOFDMs and extended them to solve singularly perturbed time-dependent partial differential equations. In this paper, we design and analyze a robust FOFDM to solve a system of coupled singularly perturbed parabolic reaction-diffusion equations. We use the backward Euler method for the semi-discretization in time. An FOFDM is then developed to solve the resulting set of boundary value problems. The proposed method is analyzed for convergence. Our method is uniformly convergent with order one and two, respectively, in time and space, with respect to the perturbation parameters. Some numerical experiments supporting the theoretical investigations are also presented.     

Multigrid and multilevel methods for quadratic spline collocation

Multigrid methods are developed and analyzed for quadratic spline collocation equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence of the two-grid method integrated with a damped Richardson relaxation scheme as smoother is shown to be faster than 1/2, independently of the step-size. The additive multilevel versions of the algorithms are also analyzed. The development of quadratic spline collocation multigrid methods is extended to two-dimensional elliptic partial differential equations. Multigrid methods for quadratic spline collocation methods are not straightforward: because the basis functions used with quadratic spline collocation are not nodal basis functions, the design of efficient restriction and extension operators is nontrivial. Experimental results, with V-cycle and full multigrid, indicate that suitably chosen multigrid iteration is a very efficient solver for the quadratic spline collocation equations. Supported by Communications and Information Technology Ontario (CITO), Canada. Supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Weighted regularization of Maxwell equations in polyhedral domains

Summary. We present a new method of regularizing time harmonic Maxwell equations by a {\bf grad}-div term adapted to the geometry of the domain. This method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions. In the presence of reentrant corners or edges, the usual regularization is known to produce wrong solutions due the non-density of smooth fields in the variational space. We get rid of this undesirable effect by the introduction of special weights inside the divergence integral. Standard finite elements can then be used for the approximation of the solution. This method proves to be numerically efficient. Received April 27, 2001 / Revised version received September 13, 2001 / Published online March 8, 2002     

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     

Adaptive filtering

Summary. In this paper, the adaptive filtering method is introduced and analysed. This method leads to robust algorithms for the solution of systems of linear equations which arise from the discretisation of partial differential equations with strongly varying coefficients. These iterative algorithms are based on the tangential frequency filtering decompositions (TFFD). During the iteration with a preliminary preconditioner, the adaptive test vector method calculates new test vectors for the TFFD. The adaptive test vector iterative method allows the combination of the tangential frequency decomposition and other iterative methods such as multi-grid. The connection with the TFFD improves the robustness of these iterative methods with respect to varying coefficients. Interface problems as well as problems with stochastically distributed properties are considered. Realistic numerical experiments confirm the efficiency of the presented algorithms. Received June 26, 1996 / Revised version received October 7, 1996     

Evolution-Galerkin methods and their supraconvergence

Summary. Evolution-Galerkin methods for partial differential equations of the form are characterised by (i) the use of some form of approximation to the corresponding evolution operator , and (ii) projection onto an approximation space to obtain . In this paper we concentrate on characteristic-Galerkin and Lagrange-Galerkin methods to derive basic error estimates for multidimensional convection problems. Methods covered include those using recovery techniques to improve accuracy. Many schemes exhibit a supraconvergence phenomenon and a general technique for its analysis is given, together with a number of particular examples. Received July 5, 1993 / Revised version received February 6, 1995     

Spline collocation for strongly elliptic equations on the torus

Summary We prove convergence and error estimates in Sobolev spaces for the collocation method with tensor product splines for strongly elliptic pseudodifferential equations on the torus. Examples of applications include elliptic partial differential equations with periodic boundary conditions but also the classical boundary integral operators of potential theory on torus-shaped domains in three or more dimensions. For odd-degree splines, we prove convergence of nodal collocation for any strongly elliptic operator. For even-degree splines and midpoint collocation, we find an additional condition for the convergence which is satisfied for the classical boundary integral operators. Our analysis is a generalization to higher dimensions of the corresponding analysis of Arnold and Wendland [4].     

Lower bounds for three-dimensional nonoverlapping domain decomposition algorithms

Summary. Lower bounds for the condition numbers of the preconditioned systems are obtained for the wire basket preconditioner and the Neumann-Neumann preconditioner in three dimensions. They show that the known upper bounds are sharp. Received January 28, 2001 / Revised version received September 3, 2001 / Published online January 30, 2002 This work was supported in part by the National Science Foundation under Grant Nos. DMS-9600133 and DMS-0074246     

Parallel domain decomposition procedures of improved D-D type for parabolic problems

Two parallel domain decomposition procedures for solving initial-boundary value problems of parabolic partial differential equations are proposed. One is the extended D-D type algorithm, which extends the explicit/implicit conservative Galerkin domain decomposition procedures, given in [5], from a rectangle domain and its decomposition that consisted of a stripe of sub-rectangles into a general domain and its general decomposition with a net-like structure. An almost optimal error estimate, without the factor H^−1/2 given in Dawson-Dupont’s error estimate, is proved. Another is the parallel domain decomposition algorithm of improved D-D type, in which an additional term is introduced to produce an approximation of an optimal error accuracy in L²-norm.     

Analysis of a finite element method for the drift-diffusion semiconductor device equations: the multidimensional case

Summary. An explicit finite element method for numerically solving the drift-diffusion semiconductor device equations in two space dimensions is analyzed. The method is based on the use of a mixed finite element method for the approximation of the electric field and a discontinuous upwinding finite element method for the approximation of the electron and hole concentrations. The mixed method gives an approximate electric field in the precise form needed by the discontinuous method, which is trivially conservative and fully parallelizable. It is proven that the method produces uniformly bounded concentrations and electric fields and that it converges to the exact solution provided there is a convergent subsequence of the electron concentrations. Numerical simulations are presented that display the performance of the method and indicate the behavior of the solution. Received September 9, 1993 / Revised version received May 25, 1994     

Domain decomposition method and elastic multi-structures: The stiffened plate problem

Summary Domain decomposition methods allow faster solution of partial differential equations in many cases. The efficiency of these methods mainly depends on the variables and operators chosen for the coupling between the subdomains; it is the preconditioning problem. In the modeling of multistructures, the partial differential equations have some specific properties that must be taken into account in a domain decomposition method. Different kinds of elliptic problems modeling stiffened plates in linearized elasticity are compared. One of them is remarkable as far as domain decomposition is concerned, since it is possible to associate particularly efficient preconditioner. A theoretical estimate for the conditioning is given, which is confirmed by several numerical experiments.     

Circulant preconditioners for second order hyperbolic equations

Linear systems arising from implicit time discretizations and finite difference space discretizations of second-order hyperbolic equations in two dimensions are considered. We propose and analyze the use of circulant preconditioners for the solution of linear systems via preconditioned iterative methods such as the conjugate gradient method. Our motivation is to exploit the fast inversion of circulant systems with the Fast Fourier Transform (FFT). For second-order hyperbolic equations with initial and Dirichlet boundary conditions, we prove that the condition number of the preconditioned system is ofO() orO(m), where is the quotient between the time and space steps andm is the number of interior gridpoints in each direction. The results are extended to parabolic equations. Numerical experiments also indicate that the preconditioned systems exhibit favorable clustering of eigenvalues that leads to a fast convergence rate.     

The splitting finite-difference time-domain methods for Maxwell's equations in two dimensions

In this paper, we consider splitting methods for Maxwell's equations in two dimensions. A new kind of splitting finite-difference time-domain methods on a staggered grid is developed. The corresponding schemes consist of only two stages for each time step, which are very simple in computation. The rigorous analysis of the schemes is given. By the energy method, it is proved that the scheme is unconditionally stable and convergent for the problems with perfectly conducting boundary conditions. Numerical dispersion analysis and numerical experiments are presented to show the efficient performance of the proposed methods. Furthermore, the methods are also applied to solve a scattering problem successfully.     

Extrapolation methods in Lie groups

Summary. The numerical solution of differential equations on Lie groups by extrapolation methods is investigated. The main principles of extrapolation for ordinary differential equations are extended on the general case of differential equations in noncommutative Lie groups. An asymptotic expansion of the global error is given. A symmetric method is given and quadratic asymptotic expansion of the global error is proved. The theoretical results are verified by numerical experiments. Received September 27, 1999 / Revised version received February 14, 2000 / Published online April 5, 2001     

Galerkin methods for nonlinear Sobolev equations

Summary We study Galerkin approximations to the solution of nonlinear Sobolev equations with homogeneous Dirichlet boundary condition in two spatial dimensions and derive optimalL ² error estimates for the continuous Crank — Nicolson and Extrapolated Crank — Nicolson approximations.     

A domain decomposition algorithm for elliptic problems in three dimensions

Summary Most domain decomposition algorithms have been developed for problems in two dimensions. One reason for this is the difficulty in devising a satisfactory, easy-to-implement, robust method of providing global communication of information for problems in three dimensions. Several methods that work well in two dimension do not perform satisfactorily in three dimensions.A new iterative substructuring algorithm for three dimensions is proposed. It is shown that the condition number of the resulting preconditioned problem is bounded independently of the number of subdomains and that the growth is quadratic in the logarithm of the number of degrees of freedom associated with a subdomain. The condition number is also bounded independently of the jumps in the coefficients of the differential equation between subdomains. The new algorithm also has more potential parallelism than the iterative substructuring methods previously proposed for problems in three dimensions.This work was supported in part by the National Science Foundation under grant NSF-CCR-8903003 and by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.     

Galerkin-wavelet methods for two-point boundary value problems

Summary Anti-derivatives of wavelets are used for the numerical solution of differential equations. Optimal error estimates are obtained in the applications to two-point boundary value problems of second order. The orthogonal property of the wavelets is used to construct efficient iterative methods for the solution of the resultant linear algebraic systems. Numerical examples are given.This work was supported by National Science Foundation     

A Zienkiewicz-type finite element applied to fourth-order problems

This paper deals with convergence analysis and applications of a Zienkiewicz-type (Z-type) triangular element, applied to fourth-order partial differential equations. For the biharmonic problem we prove the order of convergence by comparison to a suitable modified Hermite triangular finite element. This method is more natural and it could be applied to the corresponding fourth-order eigenvalue problem. We also propose a simple postprocessing method which improves the order of convergence of finite element eigenpairs. Thus, an a posteriori analysis is presented by means of different triangular elements. Some computational aspects are discussed and numerical examples are given.     

Operator adapted spectral element methods I: harmonic and generalized harmonic polynomials

Summary. The paper presents results on the approximation of functions which solve an elliptic differential equation by operator adapted systems of functions. Compared with standard polynomials, these operator adapted systems have superior local approximation properties. First, the case of Laplace's equation and harmonic polynomials as operator adapted functions is analyzed and rates of convergence in a Sobolev space setting are given for the approximation with harmonic polynomials. Special attention is paid to the approximation of singular functions that arise typically in corners. These results for harmonic polynomials are extended to general elliptic equations with analytic coefficients by means of the theory of Bergman and Vekua; the approximation results for Laplace's equation hold true verbatim, if harmonic polynomials are replaced with generalized harmonic polynomials. The Partition of Unity Method is used in a numerical example to construct an operator adapted spectral method for Laplace's equation that is based on approximating with harmonic polynomials locally. Received May 26, 1997 / Revised version received September 21, 1998 / Published online September 7, 1999