Adaptive Discontinuous Galerkin Methods of Transient Partial Differential Equations

Virginia Polytechnic Institute And State University, Blacksburg VA

Investigators

Abstract

The investigators develop very efficient adaptive discontinuous Galerkin (DG) methods for transient hyperbolic and singularly perturbed convection-diffusion problems. These methods use discontinuous bases and thus are very effective at capturing discontinuities. In addition, they simplify adaptive h-, p-, and hp-refinement, are simple to implement on unstructured meshes with irregular boundaries, parallelize very well, and satisfy local (element level) conservation principles. Adaptive enrichment by h-, p-, and r-refinement or combinations thereof has typically been guided by a posteriori estimates of discretization errors. The investigators study asymptotically correct error estimates of discretization errors under h- or p-refinement that provide very reliable measures of solution accuracy. A posteriori error estimates are developed for linear and nonlinear hyperbolic and singularly perturbed convection-diffusion problems. The investigators implement these DG methods for large-scale problems on modern parallel systems ranging from advanced supercomputers to clusters of workstations. Parallel procedures must address heterogeneity at the processor, memory, and communications levels as adaptivity greatly complicates matters. A balanced parallel computation may cease to remain so under adaptive h- or p-refinement. Computational loads must dynamically be redistributed during the solution process to restore and maintain balance. Procedures to migrate work between the processors so as to minimize the computational time have hardly been addressed in such a heterogeneous environment. Complex realistic computer simulations of physical problems like the flow about a vehicle, weather prediction, and materials processing require long times on the fastest supercomputers. The investigators develop very efficient and reliable computational techniques based on the discontinuous Galerkin method that should have advantages for parallel and network computation. Accuracy and reliability of the results are guided by estimates of solution errors that are automatically obtained as part of the computation. These are used to create adaptive procedures that assign computational resources to regions where they are needed most. The efficiency provided by adaptivity leads to more accurate computer simulations, better products, and shorter design cycles.