An Adaptive Time Step Procedure for a Parabolic Problem with Blow-up

In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blow-up of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme. Received October 4, 2001; revised March 27, 2002 Published online: July 8, 2002     

Discrete maximum principle for parabolic problems solved by prismatic finite elements

In this paper we analyze the discrete maximum principle (DMP) for a non-stationary diffusion–reaction problem solved by means of prismatic finite elements and θ$\theta$-method. We derive geometric conditions on the shape parameters of prismatic partitions and time-steps which a priori guarantee validity of the DMP. The presented numerical tests illustrate the sharpness of the obtained conditions.     

Error Estimates of a FEM with Lumping for Parabolic PDEs

L ²-norm. This well-known FEM is given by the use of the vertical line method and conforming linear finite elements on a triangulation. The main result of the paper are new estimates in the L ²-norm for the additional error term originated by lumping. Using these ones, for the FEM with lumping we can apply directly the proof technique of error estimates known for conforming FEMs. Received May 17, 2001; revised November 2, 2001 Published online February 18, 2002     

A finite volume scheme for cardiac propagation in media with isotropic conductivities

A finite volume method for solving the monodomain and bidomain models for the electrical activity of myocardial tissue is presented. These models consist of a parabolic PDE and a system of a parabolic and an elliptic PDE, respectively, for certain electric potentials, coupled to an ODE for the gating variable. The existence and uniqueness of the approximate solution is proved, and it is also shown that the scheme converges to the corresponding weak solutions for the monodomain model, and for the bidomain model when considering diagonal conductivity tensors. Numerical examples in two and three space dimensions are provided, indicating experimental rates of convergence slightly above first order for both models.     

High-Order Time-Accurate Parallel Schemes for Parabolic Singularly Perturbed Problems with Convection

The first boundary value problem for a singularly perturbed parabolic equation of convection-diffusion type on an interval is studied. For the approximation of the boundary value problem we use earlier developed finite difference schemes, ɛ-uniformly of a high order of accuracy with respect to time, based on defect correction. New in this paper is the introduction of a partitioning of the domain for these ɛ-uniform schemes. We determine the conditions under which the difference schemes, applied independently on subdomains may accelerate (ɛ-uniformly) the solution of the boundary value problem without losing the accuracy of the original schemes. Hence, the simultaneous solution on subdomains can in principle be used for parallelization of the computational method. Received December 3, 1999; revised April 20, 2000     

Variational Methods on the Space of Functions of Bounded Hessian for Convexification and Denoising

In this paper we investigate variational principles on the space of functions of bounded Hessian for denoising, for numerical calculation of convex envelopes and for approximation by convex functions.     

Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method II: Two dimensional case

A class of fifth-order weighted essentially non-oscillatory (WENO) schemes based on Hermite polynomials, termed HWENO (Hermite WENO) schemes, for solving one dimensional non-linear hyperbolic conservation law systems, was developed and applied as limiters for the Runge-Kutta discontinuous Galerkin (RKDG) methods in [J. Comput. Phys. 193 (2003) 115]. In this paper, we extend the method to solve two dimensional non-linear hyperbolic conservation law systems. The emphasis is again on the application of such HWENO finite volume methodology as limiters for RKDG methods to maintain compactness of RKDG methods. Numerical experiments for two dimensional Burgers’ equation and Euler equations of compressible gas dynamics are presented to show the effectiveness of these methods.     

Regularization methods for a problem of analytic continuation

In this paper, we prove a sharp stability estimate for the problem of analytic continuation. Based on the obtained stability estimate, a generalized Tikhonov regularization is provided and the corresponding error estimate is obtained. Moreover, we give many other regularization methods. For illustration, a numerical experiment is constructed to demonstrate the feasibility and efficiency of the proposed method.     

Reaction diffusion equations and the chronification of liver infections

A predator-prey system is used to model the time-dependent virus and lymphocyte population during a liver infection. We show mathematically that the resulting reaction-diffusion equation has non-trivial stationary solutions whenever the underlying domain is sufficiently large or fissured. The non-trivial stationary solutions are interpreted as chronic liver infections. Thus qualitative differences between acute and chronic hepatitis infections become dispensable. Finally, numerical simulations for the chronification are presented.     

Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM

In this paper we present a new automatic adaptivity algorithm for the hp-FEM which is based on arbitrary-level hanging nodes and local element projections. The method is very simple to implement compared to other existing hp-adaptive strategies, while its performance is comparable or superior. This is demonstrated on several numerical examples which include the L-shape domain problem, a problem with internal layer, and the Girkmann problem of linear elasticity. With appropriate simplifications, the proposed technique can be applied to standard lower-order and spectral finite element methods.     

A Hermitian Box-scheme for One-dimensional Elliptic Equations – Application to Problems with High Contrasts in the Ellipticity

We introduce a new box-scheme, called ``hermitian box-scheme' on the model of the one-dimensional Poisson problem. The scheme combines features of the box-scheme of Keller, [20], [13], with the hermitian approximation of the gradient on a compact stencil, which is characteristic of compact schemes, [9], [21]. The resulting scheme is proved to be 4th order accurate for the primitive unknown u and its gradient p. The proved convergence rate is 1.5 for (u,p) in the discrete L ² norm. The connection with a non standard mixed finite element method is given. Finally, numerical results are displayed on pertinent 1-D elliptic problems with high contrasts in the ellipticity, showing in practice convergence rates ranging from 1 to 2.5 in the discrete H ¹ norm. This work has been performed with the support of the GDR MOMAS, (ANDRA, CEA, EDF, BRGM and CNRS): Modélisation pour le stockage des déchets radioactifs. The author thanks especially A. Bourgeat for his encouragements and his interest in this work.     

A Note on the Energy Norm for a Singularly Perturbed Model Problem

This paper considers a singularly perturbed reaction diffusion problem. It is investigated whether adaptive approaches are successful to design robust solution procedures. A key ingredient is the a posteriori error estimator. Since robust and mathematically analysed error estimation is possible in the energy norm, the focus is on this choice of norm and its implications. The numerical performance for several model problems confirms that the proposed adaptive algorithm (in conjunction with an energy norm error estimator) produces optimal results. Hence the energy norm is suitable for the purpose considered here. The investigations also provide valuable justification for forthcoming research. Received October 25, 2001; revised July 12, 2002 Published online: October 24, 2002     

A Solution Method for a Certain Nonlinear Interface Problem in Unbounded Domains

In this paper, we consider a kind of nonlinear interface problem in unbounded domains. To solve this problem, we discuss a new coupling of finite element and boundary element by adding an auxiliary circle. We first derive the optimal error estimate of finite element approximation to the coupled FEM-BEM problem. Then we introduce a preconditioning steepest descent method for solving the discrete system by constructing a cheap domain decomposition preconditioner. Moreover, we give a complete analysis to the convergence speed of this iterative method. Received March 30, 2000; revised November 29, 2000     

A variational approximation method for 2nd order elliptic eigenvalue problems in a composite structure with nonstandard boundary conditions

In this paper we deal with the finite element analysis of a class of eigenvalue problems (EVPs) in a composite structure in the plane, consisting of rectangular subdomains which enclose an intermediate region. Nonlocal boundary conditions (BCs) of Robin type are imposed on the inner boundaries, i.e. on the interfaces of the respective subdomains with the intermediate region. On the eventual interfaces between two subdomains we impose discontinuous transition conditions (TCs). Finally, we have classical local BCs at the outer boundaries. Such problems are related to some heat transfer problems e.g. in a horizontal cross section of a wall enclosing an air cave.     

A Posteriori Error Estimation for the Dirichlet Problem with Account of the Error in the Approximation of Boundary Conditions

H ¹, independently of the discretization method chosen. In particular, our error estimator can be applied also to problems and discretizations where the Galerkin orthogonality is not available. We will present different strategies for the evaluation of the error estimator. Only one constant appears in its definition which is the one from Friedrichs' inequality; that constant depends solely on the domain geometry, and the estimator is quite non-sensitive to the error in the constant evaluation. Finally, we show how accurately the estimator captures the local error distribution, thus, creating a base for a justified adaptivity of an approximation. Received April 15, 2002; revised March 10, 2003 Published online: June 23, 2003