1. 1 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu MA557/MA578/CS557 Lecture 31

2. 2 HW9 Brief How-To Build the system matrices

3. 3 Set Up Storage for C,sigC,dCdx,dCdy…

4. 4 Compute Field Jumps dC and Derivatives

5. 5 Set Boundary Conditions And Build Residuals

6. 6 Finish up Runge-Kutta Loop and Plot Results

7. 7 Solving Elliptic Problems With DG Introduction. Suppose we wish to solve the interior Helmholtz problem, given a real lambda:

8. 8 Introducing a Mixed Method We have now amassed significant experience using DG derivative operators for first order equations. We can reuse these operators by introducing independent auxiliary variables (qx,qy) to represent the gradient of the solution using a mixed method:

9. 9 Local Discontinuous Galerkin Assuming zero Dirichlet boundary conditions at all boundaries we obtain the following LDG scheme:

10. 10 Definitions of DG Derivatives

11. 11 Properties of the DG Derivative Operators

12. 12 Summary of DG Derivative Operators

13. 13 LDG In DG Operator Notation

14. 14 Eliminating the Auxiliary Variables: We can remove those qx and qy variables: +

15. 15 Details of Auxiliary Variable Removal 1) Last equation in LDG scheme 2) Split inner-product by linearity of inner-product 3) Use property of derivative operator just defined 4) Use definition of qx and qy from first two LDG equations – noting that (D^N_x)phi and (D^N_y)phi are both in the Pp(T) space.

16. 16 Example Solution (-1,-1) (1,-1) (-1,1)(1,1) x y Set:

17. 17 Solving the System We can compute the right hand side of: We need to find piecewise polynomial u so that the above is satisfied for all piecewise polynomial phi. We will discretize the piecewise polynomial spaces. Regardless of the choice of basis – this is a symmetric, linear operator.

18. 18 Solving the System We can compute the right hand side of: We are going to use an iterative method to solve the following symmetric matrix problem to find u: Code …

19. 19 Driver

20. 20 Solver Set Up

21. 21 Action of Helmholtz Operator on x

22. 22 In Action