Hyperbolic Partial Differential Equations

In contrast to the great success of variational methods, especially finite element methods, for elliptic and even parabolic problems, their application to hyperbolic partial differential equations (PDEs) has met with somewhat less spectacular success. It is fair to say that even today, the most advanced finite element methods for hyperbolic PDEs are not completely satisfactory when compared with specialized finite volume and finite difference schemes. For example, computing monotone finite element solutions that capture solution discontinuities over a narrow band of cells remains a challenging and, for the most part, unresolved task.

The status of least-squares finite element methods (LSFEMs) for hyperbolic problems to a large degree mirrors this situation. Although the idea of replacing a hyperbolic PDE by an attractive Rayleigh–Ritz-like formulation¹ is very appealing, its straightforward application, without proper accounting for the distinctions between elliptic and hyperbolic PDEs, may lead to less than satisfactory methods.