Superconvergent Hybridizable Discontinuous Galerkin and Mixed Methods for Partial Differential Equations

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Computer simulation of physical phenomena is a highly valued tool of practical importance in a wide variety of applications in science and engineering. This project studies 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 (incompressible fluid flow, subsonic and supersonic flow) as well as to civil engineering (solid structures). This research project aims to introduce a systematic way of obtaining new, competitive discontinuous Galerkin and mixed methods that superconverge on unstructured meshes made of elements of arbitrary shape. This will be done for a wide variety of partial differential equations arising in fluid dynamics (including the incompressible Navier-Stokes equations) and continuum mechanics (including the equations of large deformation elasticity), both linear and nonlinear. The project will also consider adjoint-recovery methods that will result in a very efficient way of obtaining more accuracy than previously thought possible from general finite element approximations. By only doubling the computational effort, the order of accuracy of the approximation will be doubled. In particular, the application of this technique to methods satisfying a Galerkin orthogonality property will result in the quadrupling of the order of accuracy.