TABLE OF CONTENTS In the FEM, an important question arises: should we prefer to use conforming finite elements or nonconforming finite elements? Conforming (i.e. continuous) FE approximations are suitable for problems with sufficiently regular solutions. However, singularly perturbed problems or nonlinear conservation laws of fluid dynamics have solutions with steep gradients or discontinuities and their approximations by conforming finite elements suffer from the Gibbs phenomenon. One way to avoid this drawback is to use a suitable stabilization such as the streamline diffusion method or Galerkin least squares method and shock capturing stabilization, treated in Sections 4.1 and 4.3. As we can see there, the streamline diffusion and shock capturing terms look quite sophisticated. Moreover, the approximation of discontinuous solutions to conservation laws by continuous functions does not seem quite natural.
From this point of view, it seems that for conservation laws with a discontinuous solution, the FV method is more suitable, because the FV approximations are discontinuous on interelement interfaces, which allows better resolution of shock waves and contact discontinuities. On the other hand, as was shown in Section 4.5, the increase of accuracy in FV schemes seems to be problematic. A combination of ideas and techniques of the FV and FE methods yields the discontinuous Galerkin finite element method (DGFEM), using advantages of both approaches and allowing schemes to be obtained with a higher order of accuracy in a natural way. As was mentioned in Section 4.1.11, the DGFEM is based on the...
