Adaptive Discontinuous Galerkin Methods of Transient Partial Differential Equations

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The investigators develop very efficient adaptivediscontinuous Galerkin (DG) methods for transient hyperbolic andsingularly perturbed convection-diffusion problems. Thesemethods use discontinuous bases and thus are very effective atcapturing discontinuities. In addition, they simplify adaptiveh-, p-, and hp-refinement, are simple to implement onunstructured meshes with irregular boundaries, parallelize verywell, and satisfy local (element level) conservation principles.Adaptive enrichment by h-, p-, and r-refinement or combinationsthereof has typically been guided by a posteriori estimates ofdiscretization errors. The investigators study asymptoticallycorrect error estimates of discretization errors under h- orp-refinement that provide very reliable measures of solutionaccuracy. A posteriori error estimates are developed for linearand nonlinear hyperbolic and singularly perturbedconvection-diffusion problems. The investigators implement theseDG methods for large-scale problems on modern parallel systemsranging from advanced supercomputers to clusters of workstations.Parallel procedures must address heterogeneity at the processor,memory, and communications levels as adaptivity greatlycomplicates matters. A balanced parallel computation may ceaseto remain so under adaptive h- or p-refinement. Computationalloads must dynamically be redistributed during the solutionprocess to restore and maintain balance. Procedures to migratework between the processors so as to minimize the computationaltime have hardly been addressed in such a heterogeneousenvironment. Complex realistic computer simulations of physical problemslike the flow about a vehicle, weather prediction, and materialsprocessing require long times on the fastest supercomputers. Theinvestigators develop very efficient and reliable computationaltechniques based on the discontinuous Galerkin method that shouldhave advantages for parallel and network computation. Accuracyand reliability of the results are guided by estimates ofsolution errors that are automatically obtained as part of thecomputation. These are used to create adaptive procedures thatassign computational resources to regions where they are neededmost. The efficiency provided by adaptivity leads to moreaccurate computer simulations, better products, and shorterdesign cycles.