Hermann Brunner

Blowup in diffusion equations: a survey

This paper deals with quasilinear reaction-diffusion equations for which a solution local in time exists. If the solution ceases to exist for some finite time, we say that it blows up. In contrast to linear equations blowup can occur even if the data are smooth and well-defined for all times. Depending on the equation either the solution or some of its derivatives become singular. We shall concentrate on those cases where the solution becomes unbounded in finite time. This can occur in quasilinear equations if the heat source is strong enough. There exist many theoretical studies on the question on the occurrence of blowup. In this paper we shall recount some of the most interesting criteria and most important methods for analyzing blowup. The asymptotic behavior of solutions near their singularities is only completely understood in the special case where the source is a power. A better knowledge would be useful also for their numerical treatment. Thus, not surprisingly, the numerical analysis of this type of problems is still at a rather early stage. The goal of this paper is to collect some of the known results and algorithms and to direct the attention to some open problems.

On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods

Particular cases of nonlinear mixed Volterra–Fredholm integral equations of the second kind arise in the mathematical modeling of the spatio-temporal development of an epidemic. This paper is concerned with the numerical solution of general integral equations of this type by continuous-time and discrete-time spline collocation methods. Its focus is on the derivation and analysis of methods of high order of convergence.

The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes

Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation.

The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels

In the first part of this paper we study the regularity properties of solutions of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically. The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points.

Numerical simulations of 2D fractional subdiffusion problems

The growing number of applications of fractional derivatives in various fields of science and engineering indicates that there is a significant demand for better mathematical algorithms for models with real objects and processes. Currently, most algorithms are designed for 1D problems due to the memory effect in fractional derivatives. In this work, the 2D fractional subdiffusion problems are solved by an algorithm that couples an adaptive time stepping and adaptive spatial basis selection approach. The proposed algorithm is also used to simulate a subdiffusion-convection equation.

Nonpolynomial Spline Collocation for Volterra Equations with Weakly Singular Kernels

Volterra integral equations of the second kind with weakly singular kernels possess, in general, solutions which are not smooth near the left endpoint of the interval of integration. Since ordinary polynomial spline collocation cannot lead to high-order convergence we introduce special nonpolynomial spline spaces which are modelled after the structure of these solutions near the point of nonsmooth behavior; collocation in these spaces will once more lead to high-order methods. Analogous results are derived for Volterra integro-differential equations with weakly singular kernels.

Implicitly linear collocation methods for nonlinear Volterra equations

Abstract Recently, Kumar and Sloan introduced a new collocation-type method for numerical solution of Hammerstein integral equations. In the present paper we apply this method (which will be referred to as the implicitly linear collocation method) to nonlinear Volterra integral and integro-differential equations and discuss its connection with the iterated collocation method.

Iterated Collocation Methods and Their Discretizations for Volterra Integral Equations

It is known that the numerical solution of Volterra integral equations of the second kind by polynomial spline collocation at the Gauss–Legendre points does not lead to local superconvergence at the knots of the approximating function. In the present paper we show that iterated collocation approximation restores optimal local superconvergence at the knots but does not yield global superconvergence on the entire interval of integration, in contrast to Fredholm integral equations with smooth kernels. We also analyze the discretized versions (obtained by suitable numerical quadrature) of the collocation and iterated collocation methods.

Finite element methods for optimal control problems governed by integral equations and integro-differential equations

In this paper, we analyze finite-element Galerkin discretizations for a class of constrained optimal control problems that are governed by Fredholm integral or integro-differential equations. The analysis focuses on the derivation of a priori error estimates and a posteriori error estimators for the approximation schemes.