Error Estimates of Optimal Order for Finite Element Methods with Interpolated Coefficients for the Nonlinear Heat Equation

Error estimates are shown for some spatially discrete Galerkin finite element methods for a nonlinear heat equation. The numerical schemes studied are based on a classical transformation of the dependent variable by means of the enthalpy and the Kirchhoff transformation, and on numerical quadrature by means of interpolation of coefficients. The results depend on superconvergence in the gradient for a related elliptic problem, which is shown to hold in two cases: (i) piecewise linears on a piecewise uniform mesh in two dimensions; (ii) piecewise polynomials of any degree in one dimension with the nodes chosen as the Lobatto points.