Analysis and Applications of the Discontinuous Galerkin Method

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In its broad outlines, the research program of the P.I. aims at thedevelopment, analysis and computer implementation of numerical methods designedto approximate the solutions of some partial differential equations (pde's)that have important applications in the fields of engineering and physics. TheDiscontinuous Galerkin method constitutes the core methodology of thiseffort. The ultimate goal of the research is to make full use of this methodto develop convergent and efficient adaptive methods designed to reduce therun time of the algorithms by finding optimal or quasi-optimal meshes. Otherefforts will be directed towards the development of domain decomposition andmultigrid algorithm for the fast solution of the resulting systems of equationson single as well as multiprocessor computers.The P.I. will develop methods and adaptive algorithms for second and fourthorder elliptic problems, the incompressible Navier-Stokes equations and theCahn-Hilliard equations and use them to simulate phenomenamodeled by these equations. Adaptive methods will also be used to simulatefinite-time blowup of nonlinear evolution equations. Some areas ofapplications are chemical reactions where the geometry of the domain plays animportant role and the simulation of tumor growth.Scientific computing is playing an increasingly important role in the progressof Science as a cost effective alternative to "real life" experiments whichcould be very costly, say wind tunnel experiments in aircraft design, or evenimpossible to duplicate, such as supernova explosions and other astrophysicalphenomena. It is worth mention that numerical simulations are playing a crucialrole in the identification of the potential effects of global warming. The U.S.government, through its various funding agencies, has made important investmentsby the creation of supercomputing centers equipped with the latestgeneration of massively parallel computers. To extract the full power of thesemachines, with some having tens of thousands of individual processors, efficientand "scalable" methods and algorithms must be developed to keep pace with theadvances in hardware. Indeed, most current algorithms fall short of harnessingthe full power of these computers especially when the number of processorsexceeds a few thousand. The Discontinuous Galerkin method is a recentlyintroduced methodology with great potential and wide applicability. Its manyattributes include flexibility, ability to handle complex geometries andscalability. Further understanding of this approach and the development ofefficient and parallel computer codes will have a positive impact on theever increasing areas of Science that make essential use of numericalsimulations. Finally, two graduate students are actively participating in thisproject as partial fulfillment of their Ph.D. degree requirements. This willachieve another goal of this project which is to contribute to the training ofthe next generation of researchers.