When finite-difference methods are used to solve the elastic wave equation in a discontinuous medium, the error has two dominant components. Dispersive errors lead to artificial wave trains. Errors from interfaces lead to circular wavefronts emanating from each location where the interface appears 'jagged' to the rectangular grid.The pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy. With this method, dispersive errors are essentially eliminated. The mappings introduced in this paper also eliminate the other dominant error source. Test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost.Although the mapping method is described here in connection with the pseudospectral method, it can also be used with high-order finite-difference approximations.