The Fourier finite element method for the corner singularity expansion of the Heat equation

Abstract

Near the non-convex vertex the solution of the Heat equation is of the form u = (c star epsilon) chi r(pi/omega) sin(pi theta/omega) + omega, omega is an element of L-2(R+; H-2), where c is the stress intensity function of the time variable t,* the convolution, epsilon (x, t) = re(-r2/4t)/2 root pi t(3), chi a cutoff function and omega the opening angle of the vertex. In this paper we use the Fourier finite element method for approximating the stress intensity function c and the regular part omega, and derive the error estimates depending on the regularities of c and omega. We give some numerical examples, confirming the derived convergence rates. (C) 2014 Elsevier Ltd. All rights reserved.