Galerkin finite element methods for wave problems

We compare here the accuracy, stability and wave propagation properties of a few Galerkin methods. The basic Galerkin methods with piecewise linear basis functions (called G1FEM here) and quadratic basis functions (called G2FEM) have been compared with the streamwise-upwind Petrov Galerkin (SUPG) method for their ability to solve wave problems. It is shown here that when the piecewise linear basis functions are replaced by quadratic polynomials, the stencils become much larger (involving five overlapping elements), with only a very small increase in spectral accuracy. It is also shown that all the three Galerkin methods have restricted ranges of wave numbers and circular frequencies over which the numerical dispersion relation matches with the physical dispersion relation — a central requirement for wave problems. The model one-dimensional convection equation is solved with a very fine uniform grid to show the above properties. With the help of discontinuous initial condition, we also investigate the Gibbs’ phenomenon for these methods.