Interpolation of coefficients and transformation of the dependent variable in finite element methods for the non‐linear heat equation

Abstract

Error estimates are shown for some spatially discrete Galerkin finite element methods for a non-linear heat equation. The approximation schemes studied are based on the introduction of the enthalpy as a new dependent variable, and also on the application of the Kirchhoff transformation and on interpolation of the non-linear coefficients into standard Lagrangian finite element spaces.

Fully Discrete Error Estimation by the Method of Lines for a Nonlinear Parabolic Problem

“…We assume that the boundary and initial conditions are consistent. If f is constant, the existence and uniqueness of u follows immediately by applying the well-known Kirchhoff transformation (see [6], [8]). For non-constant f we can use the concept of pseudomonotone operators.…”

Superconvergence of a discontinuous finite element method for a nonlinear ordinary differential equation

“…The finite element method with interpolated coefficients was introduced and analyzed for semilinear parabolic problems in Zlamal et al [13]. Later Larsson-Thomee-Zhang studied the semidiscrete linear triangular finite element and obtained an error estimate [5]. Chen-Larsson-Zhang derived almost optimal order convergence on piecewise uniform triangular meshes by use of superconvergence techniques [2].…”

P 1-Nonconforming Quadrilateral Finite Volume Methods for the Semilinear Elliptic Equations

“…In 1980, this method was introduced and analyzed firstly for the semilinear parabolic problems by Zlamal [46]. Later, Larson, Thomee and Zhang [22] studied the semidiscrete linear triangular FEM with interpolated coefficients, and Chen, Larson and Zhang [8] derived almost optimal order convergence on a uniform triangular mesh by using the piecewise linear finite element space and superconvergence techniques. Xiong and Chen [37][38][39] studied the superconvergence of triangular quadratic FEM for the nonlinear ordinary differential equation and the Q 1 -conforming rectangular FEM with interpolated coefficients for the semilinear elliptic problem, respectively.…”