Pollution Errors in the Wave Equation

Dissipation is known as the phenomenon that a wave loses its mechanical energy as time passes, and dispersion is that each Fourier mode exp(i(kx+wt)) has different speed, where k is the real wave number and w is the frequency. In this respect, the wave equatio nis neither dissipative nor dispersive, so it would be natural for us to expect that an initial function propagates or travels without any deformation as time passes. However, when some finite element methods are applied to the wave equation (or Helmholtz equation), its numerical solution often does not follow our expectation. These numerical errors certainly occur when the initial condition contains a Fourier mode with a large wave number k or a high frequency w. They are called pollution errors, of which there are two main types: numerical dissipation and dispersion errors.

The pollution errors can be viewed as how small the mesh size h must be with respect to k. More exactly, any numerical method must satisfy kh<1 for a small relative error, and k^2h must be small for quasioptimality of the standard Galerkin solution. It is known that a small mesh size can reduce the pollution errors, but the small mesh depending on k would cause a huge linear system of equations. Hence, my research is focused on analyzing the pollution errors mathematically and to find effective numerical methods to reduce the errors without using very small h. In the one-dimensional case, generalized finite element methods (GFEMs) without pollution errors (Babuška et al., 1995) were introduced. On the other hand, pollution errors of FEMs are inevitable in two-dimensional Helmholtz problems. My current research can be narrowed to bringing the idea of GFEMs into the VEM framework because VEMs work with non-polynomial spaces. We expect that this research will contribute to reducing the pollution errors in two or more dimensions.