Event № 1072
High-order accurate methods for the numerical solution of partial differential equations promise higher accuracy per degree of freedom, and therefore more predictive results when compared with traditional low-order methods. However, these methods can also present some new challenges, both in terms of robustness and efficiency.For example, high-order methods for hyperbolic conservation laws often do not naturally satisfy maximum principles (or, more generally, convex invariants). This can result in spurious oscillations, nonphysical solutions, or other issues such as negative density or pressure. Additionally, high-order methods are often computationally expensive, and their use on large-scale problems can be impractical or infeasible.In this talk, I will describe recent developments improving the robustness and efficiency of high-order finite element methods. These include high-order invariant domain preserving discontinuous Galerkin methods for hyperbolic conservation laws, and uniformly convergent matrix-free linear solvers for elliptic problems.