The spectral method is proposed for the partial integro-differential equations with a weakly singular kernel on irregular domains. The space discretization is based on the nodal spectral element method using the Lagrange polynomials basis associated with the Gauss–Lobatto–Legendre quadrature nodes. Also the model is discretized in time with the Legendre spectral Galerkin method. The discretization leads to conversion of the problem to a Sylvester matrix equation which can be solved efficiently by the QZ algorithm (Gardiner et al., 1992). The convergence of the method is proven by providing a priori L 2 -error estimate. Numerical results illustrate the efficiency and spectral accuracy of the proposed method.

Space–time spectral method for a weakly singular parabolic partial integro-differential equation on irregular domains

Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000

Error estimates and adaptive finite element methods: A bibliography (1990‐2000)

Estimates of variable stepsize Runge--Kutta methods for sectorial evolution equations with nonsmooth data

Fully spectral collocation method for nonlinear parabolic partial integro-differential equations

We consider a Cauchy problem for the sectorial evolution equation with generally variable operator in a Banach space. Variable stepsize discretizations of this problem by means of a strongly A(φ)-stable Runge-Kutta method are studied. The stability and error estimates of the discrete solutions are derived for wider families of nonuniform grids than quasiuniform ones (in particular, if the operator in question is constant or Lipschitz-continuous, for arbitrary grids).

On variable stepsize Runge-Kutta approximations of a Cauchy problem for the evolution equation

The upper bounds of variable stepsize approximations of holomorphic semigroups are derived.

On the bounds of approximations of holomorphic semigroups

The maximum norm error estimates of the Galerkin finite element approximations to the solutions of differential and integro-differential multi-dimensional parabolic problems are considered. Our method is based on the use of the discrete version of the elliptic-Sobolev inequality and some operator representations of the finite element solutions. The results of the present paper lead to the error estimates of optimal or almost optimal order for the case of simplicial Lagrangian piecewise polynomial elements.

N. Yu. Bakaev

Papers

On the bounds of approximations of holomorphic semigroups

On the Galerkin finite element approximations to multi-dimensional differential and integro-differential parabolic equations