A multiresolution and smooth fictitious domain method for one-dimensional elliptical and Stefan problems

We present a wavelet based multiresolution method coupled with smooth fictitious domain and wavelet–vaguelette method to solve the one-dimensional elliptic problem equipped with Dirichlet boundary condition. Its advantages over the classical fictitious domain method are analyzed by interior error estimate and numerical examples. The efficiency of our method is pointed out by considering a Stefan problem with exact solution in one dimension.