These researchers are studying a posteriori error estimates for numerical methods for convection-diffusion problems. A posteriori estimates are important when adapting the finite element mesh and time-stepping strategy so as to control errors in the solution while minimizing the amount of computational effort required.
With the a posteriori error estimates, it is possible to devise mathematically sound hp-adaptive strategies. This research is devising practical adaptive strategies for the discontinuous Galerkin (DG) schemes developed in the last several years. These methods are ideal for convection-diffusion equations since they are robust, high-order accurate and since they allow for easy refinement. They are also highly parallelizable since they use discontinuous approximations. This feature is strongly exploited in the a posteriori error estimates.
Samuel Albert, Graduate Student Researcher
Paul E. Castillo, Graduate Student Researcher
Ilaria Perugia, Research Associate
Dominik Schoetzau, Research Associate
Xiang-Rong Yang, Graduate Student Researcher
99/219 |
"The Local Discontinuous Galerkin Method for Contaminant Transport," V. Aizinger, C. Dawson, B. Cockburn, and P. Castillo, University of Minnesota Supercomputing Institute Research Report UMSI 99/219, November 1999. Submitted for publication. |
99/220 |
"The Development of Discontinuous Galerkin Methods," B. Cockburn, G.E. Karniadakis, and C.-W. Shu, University of Minnesota Supercomputing Institute Research Report UMSI 99/220, November 1999. |
Applications in mind include the equations of shallow water flow, chemically reactive transport in groundwater and surface water, and the modeling of brain tumor cell growth and treatment. These problems, though varied, share common characteristics including moving fronts, strong nonlinear effects, transient behavior, and a need for long-time integration. The DG methods are well-suited for approximating them.
These researchers have developed a code to implement the DG method for purely convective problems. The diffusive part code is now being developed. The a posteriori error estimates are being tested in simple problems-linear convection and linear convection-diffusion. Although these test problems are simple, the problem is mathematically very complicated and intense experimentation is necessary.
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URL: http://www.msi.umn.edu/about/publications/annualreport/ar2000/depts/IT/Math/cockburn.html |
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