Many physical phenomena are represented by systems of partial differential equations having solutions with local and nonuniform multi-scale behavior such as moving fronts and boundary layers observed in fluid dynamics, combustion, biology, and materials science processes. Since these nonuniformities may form, evolve, and disappear with time, standard numerical methods become either very inefficient or impossible to use. However, Adaptive methods which variable mesh sizes and orders of approximation in both time and space are very efficient in resolving local nonuniform behaviors.
Towards this, we are developing dynamic, efficient, robust, and accurate methods that handle nonuniform behaviors. These algorithms use information available during the computations to improve the quality of the solution and are guided by error estimates which indicate that elemental discretization error decrease by either decreasing the element size h or h-refinement, by increasing the polynomial degree p or p-refinement, or by repositioning the finite element vertices to equilibrate the mesh. Our software combines these enrichment techniques to obtain optimal algorithms for various time dependent systems. As mentioned above, in the heart of these algorithms lie some error estimators or indicators. Usually, error estimators or indicators are obtained by solving local problems using an approximation of degree higher that the finite element solution. We continue to work in this area and develop more efficient and reliable error estimators that can be used to control the accuracy of the solution. Furthermore, Sharp layers and fronts usually appear in the solution of singularly perturbed and nonlinear problems. Therefore, developing a posteriori error estimates for these classes of problems is more useful and challenging. We will also continue to work on stable higher order finite elements methods for singularly perturbed elliptic and parabolic systems in two and three dimensions.
Over the last 15 years I have been working on discontinuous Galerkin methods for differential equations. We investigate the superconvergence behavior of these methods for both diffusion and hyperbolic problems. Guided by our local error analysis we are able to construct asymptotically exact and efficient a posteriori error estimates. These error estimates have been applied to solve hyperbolic problems with adaptive DG methods