Speaker：Prof. Weiwei Sun（孙伟伟教授）
The talk focuses on optimal error estimates of FEMs for problems involving multi-physics fields, which are often described by nonlinear and strongly coupled parabolic/elliptic systems A question to be concerned is the optimality of numerical approximations for each components involved in the physical system. For many popular models, existing analysis may not be optimal for certain component. A typical example given in this talk is the incompressible miscible flow in porous media which has been widely used in many engineering areas, such as reservoir simulations and surface contaminant transport and remediation. The analysis done in the last several decades shows that classical Galerkin FEMs provide the numerical concentration of the accuracy O(hr+1 c +hs p) in L2-norm, where hc and hp denotes the meshsize for the approximation to concentration and pressure, respectively. This analysis suggests to use a higher order finite element approximation to the pressure than that to the concentration in numerical simulations to achieve the best rate of convergence. But this is misleading since the error estimate is not optimal. In this talk, we introduce our recent work on new analysis of Galerkin-Galerkin methods to establish the optimal L2 error estimate O(hr+1 + hs+1) from which one can see that the best convergence rate can be achieved by taking the same order (r = s) approximation to the concentration and pressure. Clearly Galerkin FEMs with r = s are less expensive in computation and easier for implementation. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis. Finally, we extend our analysis to Galerkin-mixed methods to obtain optimal error estimates for each components.
