An efficient method based on least-squares technique for interface problems

Abstract

In this paper, we propose and analyze a new method based on the least-squares technique in a piecewise kth order (k=3,4,5) polynomial function space for one-dimensional interface problems. By defining a residual functional, we derive a new bilinear form and the minimum residual method can be established in an alternative way. The stability of the proposed method is proven. For error estimation, we discuss the optimal convergence orders under the ‖⋅‖a-norm for general non-uniform meshes. The superconvergence behaviors of the presented method have been also discovered: the convergence rate of the element averages under the L2 and ‖⋅‖a-norms can achieve 2k−2. Our theoretical findings are verified by several numerical experiments.