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ºtÁ¿¥DÃD¡G2015 NCTS/CMMSC Seminar on Scientific Computing with Applications

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Talk 1¡GAccurate Gradient Computation for Interface Problems with Variable Coefficients

A new numerical method is proposed for interface problems with piecewise variable but discontinuous coefficient. The main motivation is not only to get second order solution but also the second order accurate the gradient for some types of interface problems. The idea is based on the fast Immersed Interface Method and second order convergence of IIM for the solution and the gradient (Beale and Layton). The key of the new method is to introduce the jump in the normal derivative of the solution as augmented variable and re-write the interface problem as a Laplacian of the solution with lower order terms near the interface. Thus we can get jump relations for send order derivatives using the augmented variable and lower order terms. The idea should be applicable for boundary value problems as well. An upwind type discretization is used for the finite difference discretization near or on the interface so that the discrete coefficient matrix is an M-matrix. A multi-grid solver is used to solver the linear system of equations and a GMRES iterative method is used to solve the augmented variable. Numerical examples and convergence proof are also provided.

Talk 2: An Augmented Method for Stokes-Darcy Coupling and Applications

A new finite difference method based on Cartesian meshes is proposed for solving the fluid structure interaction between a fluid flow modeled by the tokes equations and a porous media modeled by the Darcy\'s law. The idea is to introduce several augmented variables along the interface between the fluid flow and the porous media so that the problem can be decoupled as several Poisson equations. The augmented variables should be chosen so that the Beavers-Joseph-Saffman (BJS) and other interface conditions are satisfied. In the discretization, the augmented variables have co-dimension one compared with that of the primitive variables and are solved through the Schur complement system. A nontrivial analytic solution with a circular interface is constructed to check second order convergency of the proposed method. Numerical examples with various interfaces and parameters are also presented. Some simulations show interesting behaviors of the fluid structure interaction between the fluid flow and the porous media. The computational framework can be applied to other multi-phase and multi-physics problems.