Pieter W. Hemker

Sparse-grid finite-volume multigrid for 3D-problems

We introduce a multigrid algorithm for the solution of a second order elliptic equation in three dimensions. For the approximation of the solution we use a partially ordered hierarchy of finite-volume discretisations. We show that there is a relation with semicoarsening and approximation by more-dimensional Haar wavelets. By taking a proper subset of all possible meshes in the hierarchy, a sparse grid finite-volume discretisation can be constructed.The multigrid algorithm consists of a simple damped point-Jacobi relaxation as the smoothing procedure, while the coarse grid correction is made by interpolation from several coarser grid levels.The combination of sparse grids and multigrid with semi-coarsening leads to a relatively small number of degrees of freedom,N, to obtain an accurate approximation, together with anO(N) method for the solution. The algorithm is symmetric with respect to the three coordinate directions and it is fit for combination with adaptive techniques.To analyse the convergence of the multigrid algorithm we develop the necessary Fourier analysis tools. All techniques, designed for 3D-problems, can also be applied for the 2D case, and — for simplicity — we apply the tools to study the convergence behaviour for the anisotropic Poisson equation for this 2D case.

A Class of Singularly Perturbed Convection-Diffusion Problems with a Moving Interior Layer. An a Posteriori Adaptive Mesh Technique

Abstract We study numerical approximations for a class of singularly perturbed convection-diffusion type problems with a moving interior layer. In a domain (segment) with a moving interface between two subdomains, we consider an initial boundary value problem for a singularly perturbed parabolic convection-diffusion equation. Convection fluxes on the subdomains are directed towards the interface. The solution of this problem has a moving transition layer in the neighbourhood of the interface. Unlike problems with a stationary layer, the solution exhibits singular behaviour also with respect to the time variable. Well-known upwind finite difference schemes for such problems do not converge ε-uniformly in the uniform norm. In the case of rectangular meshes which are (a priori or a posteriori ) locally condensed in the transition layer. However, the condition for convergence can be considerably weakened if we take the geometry of the layer into account, i.e., if we introduce a new coordinate system which captures the interface. For the problem in such a coordinate system, one can use either an a priori, or an a posteriori adaptive mesh technique. Here we construct a scheme on a posteriori adaptive meshes (based on the solution gradient), whose solution converges ‘almost ε-uniformly’.

High-order time-accurate parallel schemes for parabolic singularly perturbed problems with convection

Abstract 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.

Application of adaptive sparse-grid technique to a model singular perturbation problem

Abstract In this paper we show how, under minimal conditions, a combination extrapolation can be introduced for an adaptive sparse grid. We apply this technique for the solution of a two-dimensional model singular perturbation problem, defined on the domain exterior of a circle.

High-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with Robin boundary conditions

Abstract The boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameter ε. The order of convergence for the known ε-uniformly convergent schemes does not exceed 1. In this paper, using a defect correction technique, we construct ε-uniformly convergent schemes of highorder time-accuracy. The efficiency of the new defect-correction schemes is confirmed by numerical experiments. A new original technigue for experimental studying of convergence orders is developed for the cases where the orders of convergence in the x-direction and in the t-direction can be substantially different.

Manifold Mapping for Multilevel Optimization

We first show the idea behind a space-mapping iteration technique for the effi- cient solution of optimization problems. Then we show how space-mapping optimization can be understood in the framework of defect correction. We observe a difference between the solution of the optimization problem and the computed space-mapping solutions. We repair this discrepancy by exploiting the correspondence with defect correction iteration and we construct the manifold-mapping algorithm, which is as efficient as the space-mapping algorithm but converges to the accurate solution.

An adaptive multigrid strategy for convection-diffusion problems

For the solution of convection-diffusion problems we present a multilevel self-adaptive mesh-refinement algorithm to resolve locally strong varying behavior, like boundary and interior layers. The method is based on discontinuous Galerkin (Baumann-Oden DG) discretization. The recursive mesh-adaptation is interwoven with the multigrid solver. The solver is based on multigrid V-cycles with damped block-Jacobi relaxation as a smoother. Grid transfer operators are chosen in agreement with the Galerkin structure of the discretization, and local grid-refinement is taken care of by the transfer of local truncation errors between overlapping parts of the grid. We propose an error indicator based on the comparison of the discrete solution on the finest grid and its restriction to the next coarser grid. It refines in regions, where this difference is too large. Several results of numerical experiments are presented which illustrate the performance of the method.

Experience with the Solution of a Finite Difference Discretization on Sparse Grids

In a recent paper [10], we described and analyzed a finite difference discretization on adaptive sparse grids in three space dimensions. In this paper, we show how the discrete equations can be efficiently solved in an iterative process. Several alternatives have been studied before in Sprengel [16], where multigrid algorithms were used. Here, we report on our experience with BiCGStab iteration. It appears that, applied to the hierarchical representation and combined with Nested Iteration in a cascadic algorithm, BiCGStab shows fast convergence, although the convergence rate is not truly independent of the meshsize.

Molecular interaction and transport limitation in macromolecular binding to surfaces.

Binding of macromolecules to surfaces, or to surface-attached binding partners, is usually described by the classical Langmuir model, which does not include interaction between incoming and adsorbed molecules or between adsorbed molecules. The present study introduces the “Surfint” model, including such interactions. Instead of the exponential binding behaviour of the Langmuir model, the Surfint model has tanh binding equations, as confirmed by a random sequential adsorption (RSA) computer simulation. For high binding affinity, sorption kinetics become diffusion-limited as described by the existing unstirred-layer model “Unstir”, for which we present the exact analytical solution of its binding equations expressed in Lambert W-functions. Low-affinity binding of thrombin on heparin, and high-affinity binding of prothrombin on phospholipid vesicles, were measured by ellipsometry and were best described by the Surfint and Unstir models, respectively.

Acceleration by Parallel Computations of Solving High-Order Time-Accurate Difference Schemes for Singularly Perturbed Convection-Diffusion Problems

For singularly perturbed convection-diffusion problems with the perturbation parameter ? multiplying the highest derivatives, we construct a scheme based on the defect correction method and its parallel variant that converge ?-uniformly with second-order accuracy in the time variable. We also give the conditions under which the parallel computation accelerates the solution process with preserving the higher-order accuracy of the original schemes.