A least squares finite element method with high degree element shape functions for one-dimensional Helmholtz equation

An application of least squares finite element method (LSFEM) to wave scattering problems governed by the one-dimensional Helmholtz equation is presented. Boundary conditions are included in the variational formulation following Cadenas and Villamizar's previous paper in Cadenas and Villamizar [C. Cadenas, V. Villamizar, Comparison of least squares FEM, mixed galerkin FEM and an implicit FDM applied to acoustic scattering, Appl. Numer. Anal. Comput. Math. 1 (2004) 128-139]. Basis functions consisting of high degree Lagrangian element shape functions are employed. By increasing the degree of the element shape functions, numerical solutions for high frequency problems can be easily obtained at low computational cost. Computational results show that the order of convergence agrees with well known a priori error estimates. The results compare favorably with those obtained from the application of a mixed Galerkin finite element method (MGFEM).