Curtin University of Technology, Australia )

equations with boundary and interior layers—from theory to applications

1月25日(星期六)上午10:00至12:00及下午2:00至4:00

1.Introduction to convection-diffusion equations and boundary/interior layers

2.The finite element formulation, stability analysis and error estimates

3.Applications to Navier-Stokes and semiconductor device equations

4.A posteriori error estimates for numerical solutions of convection-diffusion equations

5.Computational aspects and numerical experiments

Many engineering and physical phenomena are governed by convection-diffusion equations with a positive (singular perturbation)parameter affecting their second derivative terms. For example, the incompressible Navier-Stokes equations and the semiconductor device equations. When the parameter is much smaller than 1, the equations become hyperbolic in nature, and so the derivatives of the solutions vary very rapidly near the boundaries or/and in small subregions of the solution domain. (These are called boundary or interior layers.) In this case conventional finite difference and element methods normally give numerical solutions with non-physical, spurious oscillations. Thus, stable and accurate schemes have long been sought.

In this lecture we will discuss a novel finite element method for the numerical solutions of singularly perturbed convection-diffusion equations. This method is based on unstructured Delaunay meshes and on a local exponential approximation to a solution. We will show that the method is stable for any value of the singular perturbation parameter. This guarantees that no spurious will appear near the layer in any case. We will also give an error estimate for the scheme which improves those from the conventional piecewise linear finite elements in the sense that it does not depend on the second order seminorm of the exact solution. Applications of this method to the incompressible Navier-Stokes equations and the semiconductor device equations will be presented.

Apart from the above finite element method we will also discuss a recently developed mesh adaption technique which can be used in conjunction with this method. This mesh adaption technique is based on idea of flux splitting and will give both local and global a posteriori error estimates in the energy norm.

Finally, we will consider the computational aspects of the methods and present some numerical results for some model problems and the incompressible Navier-Stokes and the semiconductor device equations.