Deferred Correction Methods for Initial Boundary Value Problems

In this paper, we consider the deferred correction principle for initial boundary value problems. The method will here be applied to the discretization in time. We obtain a method of even order p by applying the implicit midpoint rule p/2 times in each time step. For the space discretization we will use a compact implicit difference scheme. We derive error estimates for the case of time dependent coefficients and present numerical experiments confirming the theoretical analysis.     

Optimal Penalty Parameters for Symmetric Discontinuous Galerkin Discretisations of the Time-Harmonic Maxwell Equations

We provide optimal parameter estimates and a priori error bounds for symmetric discontinuous Galerkin (DG) discretisations of the second-order indefinite time-harmonic Maxwell equations. More specifically, we consider two variations of symmetric DG methods: the interior penalty DG (IP-DG) method and one that makes use of the local lifting operator in the flux formulation. As a novelty, our parameter estimates and error bounds are (i) valid in the pre-asymptotic regime; (ii) solely depend on the geometry and the polynomial order; and (iii) are free of unspecified constants. Such estimates are particularly important in three-dimensional (3D) simulations because in practice many 3D computations occur in the pre-asymptotic regime. Therefore, it is vital that our numerical experiments that accompany the theoretical results are also in 3D. They are carried out on tetrahedral meshes with high-order (p=1, 2, 3, 4) hierarchic H(curl)-conforming polynomial basis functions.     

A priori error estimates in the finite element method for nonself-adjoint elliptic and parabolic interface problems

Abstract We derive a priori error estimates in the finite element method for nonselfadjoint elliptic and parabolic interface problems in a two-dimensional convex polygonal domain. Optimal H ¹-norm and sub-optimal L ²-norm error estimates are obtained for elliptic interface problems. For parabolic interface problems, the continuous-time Galerkin method is analyzed and an optimal order error estimate in the L ²(0,T;H ¹)-norm is established. Further, a discrete-in-time discontinuous Galerkin method is discussed and a related optimal error estimate is obtained. Keywords: Elliptic and parabolic interface problems, finite element method, spatially discrete scheme, discontinuous Galerkin method, error estimates Mathematics Subject Classification (1991): 65N15, 65N20     

A discontinuous Galerkin method for the wave equation

We present a new discontinuous Galerkin method for solving the second-order wave equation using the standard continuous finite element method in space and a discontinuous method in time directly applied to second-order ode systems. We prove several optimal a priori error estimates in space–time norms for this new method and show that it can be more efficient than existing methods. We also write the leading term of the local discretization error in terms of Lobatto polynomials in space and Jacobi polynomials in time which leads to superconvergence points on each space–time cell. We discuss how to apply our results to construct efficient and asymptotically exact a posteriori estimates for space–time discretization errors. Numerical results are in agreement with theory.     

Minimal Stabilization for Discontinuous Galerkin Finite Element Methods for Hyperbolic Problems

We consider a discontinuous Galerkin finite element method for the advection–reaction equation in two space–dimensions. For polynomial approximation spaces of degree greater than or equal to two on triangles we propose a method where stability is obtained by a penalization of only the upper portion of the polynomial spectrum of the jump of the solution over element edges. We prove stability in the standard h-weighted graphnorm and obtain optimal order error estimates with respect to mesh-size. The second author was supported by the Swiss National Science Foundation.     

WEB-Spline-based mesh-free finite element approximation for p-Laplacian

In this paper, we discuss the approximation of p-Laplace problem using WEB-Spline based mesh free finite elements. Along with usual weak formulation, we also consider the mixed formulation of the p-Laplace problem. We give existence, uniqueness results for both continuous and discrete problems. We also provide a priori error estimates for both the formulations.     

Uniform Pointwise Convergence of Difference Schemes for Convection-Diffusion Problems on Layer-Adapted Meshes

We consider two convection-diffusion boundary value problems in conservative form: for an ordinary differential equation and for a parabolic equation. Both the problems are discretized using a four-point second-order upwind space difference operator on arbitrary and layer-adapted space meshes. We give ɛ-uniform maximum norm error estimates O(N ⁻²ln² N(+τ)) and O(N ⁻²(+τ)), respectively, for the Shishkin and Bakhvalov space meshes, where N is the space meshnodes number, τ is the time meshinterval. The smoothness condition for the Bakhvalov mesh is replaced by a weaker condition. Received December 14, 1999; revised September 13, 2000     

A Numerical Method to Verify the Elliptic Eigenvalue Problems Including a Uniqueness Property

We propose a numerical method to enclose the eigenvalues and eigenfunctions of second-order elliptic operators with local uniqueness. We numerically construct a set containing eigenpairs which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space by using a finite element approximation and constructive error estimates. We then prove the local uniqueness separately of eigenvalues and eigenfunctions. This local uniqueness assures the simplicity of the eigenvalue. Numerical examples are presented. Received: November 2, 1998; revised June 5, 1999     

On the fast and fully discretized solution¶of integral and pseudo-differential equations¶on smooth curves

For the fast numerical solution of a fully discrete variant of the trigonometric Galerkin equations associated with periodic integral equations, we consider approximations with small residuals and provide order-optimal estimates for the associated error. The CGNR method is considered as a method with a simple iteration scheme where these approximations can be obtained by a total number of ℳ(N log N ) arithmetical operations, with N denoting the dimension of the space of trigonometric trial polynomials associated with the Galerkin method. Noise in the model of the problem as well as in the right-hand side is admitted. Received: August 1999 / Revised version: July 2000     

A posteriori error estimates for space-time finite element approximation of quasistatic hereditary linear viscoelasticity problems

We give a space-time Galerkin finite element discretisation of the quasistatic compressible linear viscoelasticity problem as described by an elliptic partial differential equation with a fading memory Volterra integral. The numerical scheme consists of a continuous Galerkin approximation in space based on piecewise polynomials of degree p>0 (cG(p)), with a discontinuous Galerkin piecewise constant (dG(0)) or linear (dG(1)) approximation in time. A posteriori Galerkin-error estimates are derived by exploiting the Galerkin framework and optimal stability estimates for a related dual backward problem. The a posteriori error estimates are quite flexible: strong L_p-energy norms of the errors are estimated using time derivatives of the residual terms when the data are smooth, while weak-energy norms are used when the data are non-smooth (in time).We also give upper bounds on the dG(0)cG(1) a posteriori error estimates which indicate optimality. However, a complete analysis is not given.     

A fully discrete local discontinuous Galerkin method for one-dimensional time-fractional Fisher's equation

In this paper, we consider the local discontinuous Galerkin (LDG) finite element method for one-dimensional time-fractional Fisher's equation, which is obtained from the standard one-dimensional Fisher's equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0<α<1). The proposed LDG is based on the LDG finite element method for space and finite difference method for time. We prove that the method is stable, and the numerical solution converges to the exact one with order O(h^k+1+τ^2?α), where h, τ and k are the space step size, time step size, polynomial degree, respectively. The numerical experiments reveal that the LDG is very effective.     

Fully reliable error control for first-order evolutionary problems

This work is focused on the application of functional-type a posteriori error estimates and corresponding indicators to a class of time-dependent problems. We consider the algorithmic part of their derivation and implementation and also discuss the numerical properties of these bounds that comply with obtained numerical results. This paper examines two different methods of solution approximation for evolutionary models, i.e., a time-marching technique and a space–time approach. The first part of the study presents an algorithm for global minimisation of the majorant on each of discretisation time-cylinders (time-slabs), the effectiveness of this approach to error estimation is confirmed by extensive numerical tests. In the second part of the publication, the application of functional error estimates is discussed with respect to a space–time approach. It is followed by a set of extensive numerical tests that demonstrates the efficiency of proposed error control method.The numerical results obtained in this paper rely on the implementation carried out using open source software, which allows formulating the problem in a weak setting. To work in a variational framework appears to be very natural for functional error estimates due to their derivation method. The search for the optimal parameters for the majorant is done by a global functional minimisation, which to the authors’ knowledge is the first work using this technique in an evolutionary framework.     

Superconvergence of Discontinuous Finite Element Solutions for Transient Convection–diffusion Problems

We present a study of the local discontinuous Galerkin method for transient convection–diffusion problems in one dimension. We show that p-degree piecewise polynomial discontinuous finite element solutions of convection-dominated problems are O(Δx ^p+2) superconvergent at Radau points. For diffusion- dominated problems, the solution’s derivative is O(Δx ^p+2) superconvergent at the roots of the derivative of Radau polynomial of degree p+1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Worst-case analysis of the least-squares method and related identification methods

We consider worst-case analysis of system identification under less restrictive assumptions on the noise than the l_∞ bounded error condition. It is shown that the least-squares method has a robust convergence property in l₂ identification, but lacks a corresponding property in l₁ identification (as well as in all other non-Hilbert space settings). The latter result is in stark contrast with typical results in asymptotic stochastic analysis of the least-squares method. Furthermore, it is shown that the Khintchine inequality is useful in the analysis of least l^p identification methods.     

An Efficient and Accurate Method for 3D-Point Reconstruction from Multiple Views

In this paper we consider the problem of finding the position of a point in space given its projections in multiple images taken by cameras with known calibration and pose. Ideally the 3D point can be obtained as the intersection of multiple known rays in space. However, with noise the rays do not meet at a single point generally. Therefore, it is necessary to find a best point of intersection. In this paper we propose a modification of the method (Ma et al., 2001. Journal of Communications in Information and Systems, (1):51–73) based on the multiple-view epipolar constraints. The solution is simple in concept and straightforward to implement. It includes generally two steps: first, image points are corrected through approximating the error model to the first order, and then the 3D point can be reconstructed from the corrected image points using any generic triangulation method. Experiments are conducted both on simulated data and on real data to test the proposed method against previous methods. It is shown that results obtained with the proposed method are consistently more accurate than those of other linear methods. When the measurement error of image points is relatively small, its results are comparable to those of maximum likelihood estimation using Newton-type optimizers; and when processing image-point correspondences cross a small number of views, the proposed method is by far more efficient than the Newton-type optimizers.     

Numerical solutions for a class of differential equations in linear viscoelasticity

We consider numerical solutions by finite element methods for a class of hyperbolic integro-differential equations in linear viscoelasticity. The kernel under consideration is assumed to be of positive type or monotonic. The semidiscrete and fully discrete (with positive discretization of the kernel) finite element methods are studied, andL ² error estimates are demonstrated for smooth data. This work is supported in part by NSERC (Canada).     

Adjoint Recovery of Superconvergent Linear Functionals from Galerkin Approximations. The One-dimensional Case

In this paper, we extend the adjoint error correction of Pierce and Giles (SIAM Rev. 42, 247–264 (2000)) for obtaining superconvergent approximations of functionals to Galerkin methods. We illustrate the technique in the framework of discontinuous Galerkin methods for ordinary differential and convection–diffusion equations in one space dimension. It is well known that approximations to linear functionals obtained by discontinuous Galerkin methods with polynomials of degree k can be proven to converge with order 2k + 1 and 2k for ordinary differential and convection–diffusion equations, respectively. In contrast, the order of convergence of the adjoint error correction method can be proven to be 4k + 1 and 4k, respectively. Since both approaches have a computational complexity of the same order, the adjoint error correction method is clearly a competitive alternative. Numerical results which confirm the theoretical predictions are presented.     

An a posteriori error estimate for a linear-nonlinear transmission problem in plane elastostatics

We consider the coupling of dual-mixed finite element and boundary element methods to solve a linear-nonlinear transmission problem in plane hyperelasticity with mixed boundary conditions. Besides the displacement and the stress tensor, we introduce the strain tensor as an additional unknown, which yields a two-fold saddle point operator equation as the corresponding variational formulation. We derive a reliable a posteriori error estimate that depends on the solution of local Dirichlet problems and on residual terms on the transmission and Neumann boundaries, which are given in a negative order Sobolev norm. Our approach does not need the exact Galerkin solution, but any reasonable approximation of it. In addition, the analysis does not depend on special finite element or boundary element subspaces. However, for certain specific subspaces we are able to provide two fully local a posteriori error estimates, in which the residual terms are bounded by weighted local L ²-norms. Further, one of the error estimates does not require the explicit solution of the local problems. Received: November 2000 / Revised version: December 2001 This research was partially supported by Fondecyt-Chile through research projects 1980122 and 2000124, and the FONDAP Program in Applied Mathematics, and by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program.     

Optimal error estimates of the penalty method for the linearized viscoelastic flows

In this article, optimal error estimates of the penalty method for the linearized viscoelastic flows equations arising in the Oldroyd model are derived. Furthermore, error estimates for the backward Euler time discretization scheme in L ² and H ¹-norms are obtained.