A numerical scheme for solving nonhomogeneous time-dependent problems

A numerical technique for solving time-dependent problems with variable coefficient governed by the heat, convection diffusion, wave, beam and telegraph equations is presented. The Sinc–Galerkin method is applied to construct the numerical solution. The method is tested on three problems and comparisons are made with the exact solutions. The numerical results demonstrate the reliability and efficiency of using the Sinc–Galerkin method to solve such problems.     

An Alternating-Direction Sinc–Galerkin method for elliptic problems

A new Alternating-Direction Sinc–Galerkin (ADSG) method is developed and contrasted with classical Sinc–Galerkin methods. It is derived from an iterative scheme for solving the Lyapunov equation that arises when a symmetric Sinc–Galerkin method is used to approximate the solution of elliptic partial differential equations. We include parameter choices (derived from numerical experiments) that simplify existing alternating-direction algorithms. We compare the new scheme to a standard method employing Gaussian elimination on a system produced using the Kronecker product and Kronecker sum, as well as to a more efficient algorithm employing matrix diagonalization. We note that the ADSG method easily outperforms Gaussian elimination on the Kronecker sum and, while competitive with matrix diagonalization, does not require the computation of eigenvalues and eigenvectors.     

A Sinc–Collocation method for the linear Fredholm integro-differential equations

A Sinc–Collocation method for solving linear integro-differential equations of the Fredholm type is discussed. The integro-differential equations are reduced to a system of algebraic equations and Q-R method is used to establish numerical procedures. The convergence rate of the method is . Numerical results are included to confirm the efficiency and accuracy of the method even in the presence of singularities and a comparison with the rationalized Haar wavelet method is made.     

On the adaptive coupling of FEM and BEM in 2–d–elasticity

Summary. This paper concerns the combination of the finite element method (FEM) and the boundary element method (BEM) using the symmetric coupling. As a model problem in two dimensions we consider the Hencky material (a certain nonlinear elastic material) in a bounded domain with Navier–Lamé differential equation in the unbounded complementary domain. Using some boundary integral operators the problem is rewritten such that the Galerkin procedure leads to a FEM/BEM coupling and quasi–optimally convergent discrete solutions. Beside this a priori information we derive an a posteriori error estimate which allows (up to a constant factor) the error control in the energy norm. Since information about the singularities of the solution is not available a priori in many situation and having in mind the goal of an automatic mesh–refinement we state adaptive algorithms for the –version of the FEM/BEM–coupling. Illustrating numerical results are included. Received April 15, 1994 / Revised version received January 8, 1996     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

Boundary integral approximation of a heat-diffusion problem in time-harmonic regime

In this paper we propose and analyse numerical methods for the approximation of the solution of Helmholtz transmission problems in the half plane. The problems we deal with arise from the study of some models in photothermal science. The solutions to the problem are represented as single layer potentials and an equivalent system of boundary integral equations is derived. We then give abstract necessary and sufficient conditions for convergence of Petrov–Galerkin discretizations of the boundary integral system and show for three different cases that these conditions are satisfied. We extend the results to other situations not related to thermal science and to non-smooth interfaces. Finally, we propose a simple full discretization of a Petrov–Galerkin scheme with periodic spline spaces and show some numerical experiments.     

A Sinc–Collocation method for the linear Fredholm integro-differential equations

A Sinc–Collocation method for solving linear integro-differential equations of the Fredholm type is discussed. The integro-differential equations are reduced to a system of algebraic equations and Q-R method is used to establish numerical procedures. The convergence rate of the method is O( e ^{- k?N} )O{\left( {e^{{ - k{\sqrt N }}} } \right)} . Numerical results are included to confirm the efficiency and accuracy of the method even in the presence of singularities and a comparison with the rationalized Haar wavelet method is made.     

Application of the Sinc method to a dynamic elasto-plastic problem

This paper presents the application of Sinc bases to simulate numerically the dynamic behavior of a one-dimensional elastoplastic problem. The numerical methods that are traditionally employed to solve elastoplastic problems include finite difference, finite element and spectral methods. However, more recently, biorthogonal wavelet bases have been used to study the dynamic response of a uniaxial elasto-plastic rod [Giovanni F. Naldi, Karsten Urban, Paolo Venini, A wavelet-Galerkin method for elastoplasticity problems, Report 181, RWTH Aachen IGPM, and Math. Modelling and Scient. Computing, vol. 10, 2000]. In this paper the Sinc–Galerkin method is used to solve the straight elasto-plastic rod problem. Due to their exponential convergence rates and their need for a relatively fewer nodal points, Sinc based methods can significantly outperform traditional numerical methods [J. Lund, K.L. Bowers, Sinc Methods for Quadrature and Differential Equations, SIAM, Philadelphia, 1992]. However, the potential of Sinc-based methods for solving elastoplasticity problems has not yet been explored. The aim of this paper is to demonstrate the possible application of Sinc methods through the numerical investigation of the unsteady one dimensional elastic-plastic rod problem.     

Convergence of a fitted finite volume method for the penalized Black–Scholes equation governing European and American Option pricing

In this paper we present an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing American and European option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. The analysis is performed within the framework of the vertical method of lines, where the spatial discretization is formulated as a Petrov–Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems. We establish the stability and an error bound for the solutions of the fully discretized system. Numerical results are presented to validate the theoretical results.     

A finite element method and variable transformations for a forward-backward heat equation

The Galerkin finite element method for the forward-backward heat equation is generalized to a wider class of equations with the use of a result on the existence and uniqueness of a weak solution to the problems. First, the theory for the Galerkin method is extended to forward-backward heat equations which contain additional convection and mass terms on an irregular domain. Second, variable transformations are constructed and applied to solve a wide class of forward-backward heat equations that leads to a substantial improvement. Third, Error estimates are presented. Finally, conducted numerical tests corroborate the obtained results. Received February 4, 1997 / Revised version received December 8, 1997     

Numerical approximations for population growth models

This paper aims to introduce a comparison of Adomian decomposition method and Sinc–Galerkin method for the solution of some mathematical population growth models. From the computational viewpoint, the comparison shows that the Adomian decomposition method is efficient and easy to use.     

The multiplicity of solutions in non-homogeneous boundary value problems

We use a method recently devised by Bolle to establish the existence of an infinite number of solutions for various non-homogeneous boundary value problems. In particular, we consider second order systems, Hamiltonian systems as well as semi-linear partial differential equations. The non-homogeneity can originate in the equation but also from the boundary conditions. The results are more satisfactory than those obtained by the standard “Perturbation from Symmetry” method that was developed – in various forms – in the early eighties by Bahri–Berestycki, Struwe and Rabinowitz. Received: 13 August 1998 / Revised version: 6 July 1999     

Hybrid Galerkin boundary elements: theory and implementation

Summary. In this paper we present a new quadrature method for computing Galerkin stiffness matrices arising from the discretisation of 3D boundary integral equations using continuous piecewise linear boundary elements. This rule takes as points some subset of the nodes of the mesh and can be used for computing non-singular Galerkin integrals corresponding to pairs of basis functions with non-intersecting supports. When this new rule is combined with standard methods for the singular Galerkin integrals we obtain a “hybrid” Galerkin method which has the same stability and asymptotic convergence properties as the true Galerkin method but a complexity more akin to that of a collocation or Nystr?m method. The method can be applied to a wide range of singular and weakly-singular first- and second-kind equations, including many for which the classical Nystr?m method is not even defined. The results apply to equations on piecewise-smooth Lipschitz boundaries, and to non-quasiuniform (but shape-regular) meshes. A by-product of the analysis is a stability theory for quadrature rules of precision 1 and 2 based on arbitrary points in the plane. Numerical experiments demonstrate that the new method realises the performance expected from the theory. Received January 22, 1998 / Revised version received May 26, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

Wavelets collocation methods for the numerical solution of elliptic BV problems

Based on collocation with Haar and Legendre wavelets, two efficient and new numerical methods are being proposed for the numerical solution of elliptic partial differential equations having oscillatory and non-oscillatory behavior. The present methods are developed in two stages. In the initial stage, they are developed for Haar wavelets. In order to obtain higher accuracy, Haar wavelets are replaced by Legendre wavelets at the second stage. A comparative analysis of the performance of Haar wavelets collocation method and Legendre wavelets collocation method is carried out. In addition to this, comparative studies of performance of Legendre wavelets collocation method and quadratic spline collocation method, and meshless methods and Sinc–Galerkin method are also done. The analysis indicates that there is a higher accuracy obtained by Legendre wavelets decomposition, which is in the form of a multi-resolution analysis of the function. The solution is first found on the coarse grid points, and then it is refined by obtaining higher accuracy with help of increasing the level of wavelets. The accurate implementation of the classical numerical methods on Neumann’s boundary conditions has been found to involve some difficulty. It has been shown here that the present methods can be easily implemented on Neumann’s boundary conditions and the results obtained are accurate; the present methods, thus, have a clear advantage over the classical numerical methods. A distinct feature of the proposed methods is their simple applicability for a variety of boundary conditions. Numerical order of convergence of the proposed methods is calculated. The results of numerical tests show better accuracy of the proposed method based on Legendre wavelets for a variety of benchmark problems.     

Front propagation in infinite cylinders. II. The sharp reaction zone limit

This paper applies the variational approach developed in part I of this work [22] to a singular limit of reaction–diffusion–advection equations which arise in combustion modeling. We first establish existence, uniqueness, monotonicity, asymptotic decay, and the associated free boundary problem for special traveling wave solutions which are minimizers of the considered variational problem in the singular limit. We then show that the speed of the minimizers of the approximating problems converges to the speed of the minimizer of the singular limit. Also, after an appropriate translation the minimizers of the approximating problems converge strongly on compacts to the minimizer of the singular limit. In addition, we obtain matching upper and lower bounds for the speed of the minimizers in the singular limit in terms of a certain area-type functional for small curvatures of the free boundary. The conclusions of the analysis are illustrated by a number of numerical examples.     

The Petrov–Galerkin method for second kind integral equations II: multiwavelet schemes

This paper continues the theme of the recent work [Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second kind integral equations, SIAM J. Numer. Anal., to appear] and further develops the Petrov–Galerkin method for Fredholm integral equations of the second kind. Specifically, we study wavelet Petrov–Galerkin schemes based on discontinuous orthogonal multiwavelets and prove that the condition number of the coefficient matrix for the linear system obtained from the wavelet Petrov–Galerkin scheme is bounded. In addition, we propose a truncation strategy which forms a basis for fast wavelet algorithms and analyze the order of convergence and computational complexity of these algorithms. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonlinear functionals of wavelet expansions – adaptive reconstruction and fast evaluation

Summary. This paper is concerned with the efficient evaluation of nonlinear expressions of wavelet expansions obtained through an adaptive process. In particular, evaluation covers here the computation of inner products of such expressions with wavelets which arise, for instance, in the context of Galerkin or Petrov Galerkin schemes for the solution of differential equations. The central objective is to develop schemes that facilitate such evaluations at a computational expense exceeding the complexity of the given expansion, i.e., the number of nonzero wavelet coefficients, as little as possible. The following issues are addressed. First, motivated by previous treatments of the subject, we discuss the type of regularity assumptions that are appropriate in this context and explain the relevance of Besov norms. The principal strategy is to relate the computation of inner products of wavelets with compositions to approximations of compositions in terms of possibly few dual wavelets. The analysis of these approximations finally leads to a concrete evaluation scheme which is shown to be in a certain sense asymptotically optimal. We conclude with a simple numerical example. Received June 25, 1998 / Revised version received June 5, 1999 / Published online April 20, 2000 –? Springer-Verlag 2000     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.     

A fully discrete numerical scheme for weighted mean curvature flow

Summary. We analyze a fully discrete numerical scheme approximating the evolution of n–dimensional graphs under anisotropic mean curvature. The highly nonlinear problem is discretized by piecewise linear finite elements in space and semi–implicitly in time. The scheme is unconditionally stable und we obtain optimal error estimates in natural norms. We also present numerical examples which confirm our theoretical results. Received October 2, 2000 / Published online July 25, 2001     

A Robin-type non-overlapping domain decomposition procedure for second order elliptic problems

This article deals with the analysis of an iterative non-overlapping domain decomposition (DD) method for elliptic problems, using Robin-type boundary condition on the inter-subdomain boundaries, which can be solved in parallel with local communications. The proposed iterative method allows us to relax the continuity condition for Lagrange multipliers on the inter-subdomain boundaries. In order to derive the corresponding discrete problem, we apply a non-conforming Galerkin method using lowest order Crouzeix–Raviart elements. The convergence of the iterative scheme is obtained by proving that the spectral radius of the matrix associated with the fixed point iterations is less than 1. Parallel computations have been carried out and the numerical experiments confirm the theoretical results established in this paper.