Discontinuous Galerkin methods for singularly perturbed problems.

摘要

Some interesting phenomena in many fields, such as fluid dynamics, physics, chemical kinetics and combustion, biology, etc., can be described by a singular perturbed PDE. The exact solution of such problem has sharp derivatives in a layer near the outflow of the boundary, so that approximations inside the layer typically perform poorly. This work is to seek an uniformly convergent numerical solution by using discontinuous Galerkin methods. Firstly, I designed the mixed LDG formulation for one one-dimensional problem. An optimal convergence rate was proved. Secondly, I extend the mixed LDG formulation to two-dimensional problem on rectangular mesh. By using the regularity of exact solution and the property of Shishkin mesh, which is a kind of layer-adaptive mesh, an optimal convergence rate was proved. The convergence rate is weakly dependent of the small parameter. Another way to seek an approximation for the gradient was also considered by using the primal formulation of LDG methods. An optimal and uniform convergence rate was proved.