Adaptive Sparse Grids for Fusion Relevant High Dimensional PDEs.

Predicting the behavior of magnetic confinement fusion devices requires the solution of high dimensional PDEs. Traditional grid- or mesh-based methods for solving such systems in a noise-free manner quickly become intractable due to the scaling of the degrees of freedom going as O(N\textasciicircum d), sometimes called "the curse of dimensionality." We are developing an arbitrarily high-order discontinuous-Galerkin finite-element solver that leverages the sparse-grid discretization whose degrees of freedom scale as O(N*log2N\textasciicircum D-1). In this paper, we employ the adaptive aspect of our solver in a study of how adaptivity in the selection rule for truncating the tensor products affects the advantages of sparse-grids for fusion relevant problems.