Unconditionally Stable Space-Time Finite Element Method for the Shallow Water Equations

We introduce the automatic variationally stable finite element (AVS-FE) method [1, 3] for the shallow water equations (SWE). The AVS-FE method uses a first order system integral formulation of the under- lying partial differential equations (PDEs) and, in the spirit of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan [2], employs the concept of optimal test functions to ensure discrete stability. The AVS-FE method distinguishes itself by using globally conforming FE trial spaces, e.g., H1(Ω) and H(div,Ω) and their broken counterparts for the test spaces. The broken topology of the test spaces allows us to compute numerical approximations of the local restrictions of the optimal test functions in a completely decoupled fashion, i.e., element-by-element. The test functions can be com- puted with sufficient numerical accuracy by using the same local p-level as applied for the trial space. The unconditional discrete stability of the method allows for straightforward implementation of transient problems in existing FE solvers such as FEniCS. Furthermore, the AVS-FE method comes with a built-in a posteriori error estimate as well as element-wise error indicators allowing us to perform mesh adaptive refinements in both space and time. The application of this method to complex physical domains requires large FE meshes leading to signif- icant computational costs. However, since the computation of optimal test functions as well as element- wise error indicators are all local, the method is an excellent candidate for parallel processing. We show numerical verifications for the SWE utilizing the built-in error indicators to drive mesh adaptive refine- ments.