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

Near the non-convex vertex the solution of the Heat equation is of the form u=(c⋆E)χrπ/ωsin(πθω)+w, w∈L2(R+;H2), where cc is the stress intensity function of the time variable tt, ⋆⋆ the convolution, E(x,t)=re−r2/4t/2πt3, χχ a cutoff function and ωω the opening angle of the vertex. In this paper we use the Fourier finite element method for approximating the stress intensity function cc and the regular part ww, and derive the error estimates depending on the regularities of cc and ww. We give some numerical examples, confirming the derived convergence rates.