Mesh adaption strategies for steady shallow water flow

The use of unstructured grids for the numerical approximation of partial differential equations of applied mathematics has the great appeal of enabling mesh adaption based on suitable error indicators of the accuracy of the solution, refining the mesh where the numerical error is large and coarsening it where the error is small. In this way it is then possible to optimize the quality of the solution for a given computational effort. We deal here with mesh adaption applied to shallow water flow. The shallow water equations are numerically approximated by a standard Galerkin finite element method, using linear elements for the elevation field and quadratic elements for the unit-width discharge field. The advancing-in-time scheme used is of fractional step type. The standard mesh refinement technique is used; movement and elimination of nodes of the initial triangulation is not allowed. Two empirical error indicators are proposed and applied here to to an ideal case of steady flow. The numerical tests show that mesh adaption is a very reliable tool for numerical simulation of shallow water steady flow. Any of the used error indicators produce numerical results that are strongly improved with respect to a uniform mesh, with only a minor increase in the computational effort