Finite element methods of the two nonlinear integro-differential equations

Abstract

Transformation of dependent variables, such as, the Kirchhoff transformation, is a classical tool for solving nonlinear partial differential equations. In Larsson et al., this approach was used in connection with the finite element method. Zlámal used it in Galerkin approximation in space, the Euler method in time, and applied it to the solution of nonlinear heat equations. Here we give justification of the methods for the two nonlinear integro-heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit Euler method in time with quadrature rules which require the storage of relatively few values of the solution. The estimate of the discretization error in the maximum norm is introduced. The results depend on superconvergence in the gradient for a related elliptic problem, which is shown to hold only for very special meshes.