Demkowicz was supported in part by the Department of Energy [National Nuclear Security Administration] under Award Number [DE-FC52-08NA28615], and by a research contract with Boeing. Gopalakrishnan was supported in part by the National Science Foundation under grant DMS-0713833. Niemi was supported in part by KAUST.
Applied Numerical Mathematics
Discontinuous functions, Numerical analysis, Diffusion processes, Matrices -- Norms, Reaction-diffusion equations
We continue our theoretical and numerical study on the Discontinuous Petrov–Galerkin method with optimal test functions in context of 1D and 2D convection-dominated diffusion problems and hp-adaptivity. With a proper choice of the norm for the test space, we prove robustness (uniform stability with respect to the diffusion parameter) and mesh-independence of the energy norm of the FE error for the 1D problem. With hp-adaptivity and a proper scaling of the norms for the test functions, we establish new limits for solving convection-dominated diffusion problems numerically: for 1D and for 2D problems. The adaptive process is fully automatic and starts with a mesh consisting of few elements only.
Demkowicz, Leszek; Gopalakrishnan, Jay; and Niemi, Antti H., "A Class of Discontinuous Petrov–Galerkin Methods. Part III: Adaptivity" (2012). Mathematics and Statistics Faculty Publications and Presentations. 45.