Weak Galerkin Methods for Second Order Elliptic Interface Problems

Details Weak Galerkin Methods for Second Order Elliptic Interface Problems Weak Galerkin Methods for Second Order Elliptic Interface Problems

2012-01-31

arxiv.org/abs/1201.6438v2

Mathematics

Numerical Analysis

24 pages, many figures from computation

Scientific paper

Weak Galerkin methods refer to general finite element methods for PDEs in which differential operators are approximated by their weak forms as distributions. Such weak forms give rise to desirable flexibilities in enforcing boundary and interface conditions. A weak Galerkin finite element method (WG-FEM) is developed in this paper for solving elliptic partial differential equations (PDEs) with discontinuous coefficients and interfaces. The paper also presents many numerical tests for validating the WG-FEM for solving second order elliptic interface problems. For such interface problems, the solution possesses a certain singularity due to the nonsmoothness of the interface. A challenge in research is to design high order numerical methods that work well for problems with low regularity in the solution. The best known numerical scheme in the literature is of order one for the solution itself in $L_\infty$ norm. It is demonstrated that the WG-FEM of lowest order is capable of delivering numerical approximations that are of order 1.75 in the usual $L_\infty$ norm for $C^1$ or Lipschitz continuous interfaces associated with a $C^1$ or $H^2$ continuous solutions. Theoretically, it is proved that high order of numerical schemes can be designed by using the WG-FEM with polynomials of high order on each element.

Affiliated with

Also associated with

No associations

Rating

Weak Galerkin Methods for Second Order Elliptic Interface Problems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Weak Galerkin Methods for Second Order Elliptic Interface Problems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Weak Galerkin Methods for Second Order Elliptic Interface Problems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-56491