An anisotropic, fully adaptive algorithm for the solution of convection-dominated equations with semi-Lagrangian schemes

We present in this paper an anisotropic, fully adaptive spatial–temporal algorithm for the solution of convection-dominated equations in the context of semi-Lagrangian schemes. We devise the algorithm within a finite element framework suitable for higher-order finite elements, and derive a newly proposed a posteriori error indicator which allows us to control the local or truncation error in the L2L2-norm at each time step. This a posteriori error is split into temporal and spatial contributions, leading us to define an optimal time step size and an optimal triangulation, respectively. As regards the spatial adaptation, anisotropic, unstructured triangular meshes are used to capture the distinctive features of the evolving discrete solution of the governing equations. For solutions exhibiting strong anisotropies, the orientation, shape and size of the mesh triangles are provided by a metric tensor valid for linear and quadratic finite elements.Finally, we show the capabilities of the algorithm, for linear and quadratic finite elements, by a series of two- and three-dimensional benchmarks taken from the literature, involving purely convective as well as convection-dominated problems.