Superconvergent Approximations by Galerkin Methods for Partial Differential Equations

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

The computer simulation of physical phenomena is a highly valued tool of practical interest in a wide variety of applications in Engineering and Physics. The investigator will study two new, promising techniques of carrying out these simulations with highly accurate and more efficient algorithms for a wide range of problems of practical interest. They include many applications to Aerospace and Mechanics (heat flow, incompressible fluid flow, subsonic and supersonic flow) as well as to Civil and Mechanical Engineering (seismic wave propagation and elastodynamics of solid structures). The investigator proposes to continue to develop a recently uncovered adjoint-recovery method which can reduce the cost of computing approximations of linear and nonlinear functionals by Galerkin methods by several orders of magnitude. The incorporation of adaptivity techniques, to deal with the varying degree of regularity of the solution, and the extension of this approach to the computation of nonlinear functionals, like the eigenvalues of a differential operator, will render the method of great practical value. The investigator will also develop new discretization techniques for partial differential equations with Hamiltonian structure. They combine superconvergent discontinuous Galerkin space discretizations with symplectic time-marching schemes and result in methods with an energy which does not drift. Such methods are important in many applications including seismic wave propagation and elastodynamics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.