Error estimates for degenerate parabolic equation

1. Error estimates for degenerate parabolic equation Yabin Fan CASA Seminar, 21.05.2008

2. Outline • Introduction • Weak solution • Regularization • Error estimates • Summary

3. Introduction 1 • Problem P in in on , an increasing function , bounded domain in Rd , Lipschtiz continuous boundary. T, a fixed finite time.

4. Introduction 2 • may be zero at some point, then will blow up. • u lacks regularity ,consider instead. • regularize , take , make larger than some positive constant.

5. Weak solution • Classical solution, hard to find, even impossible. • Weak solution : u is called a weak solution of problem P iff , , And for all the following equation holds true.

6. Regularization 1 • In this talk, we give some assumptions about the problem P (A1). is Lipschitz and differentiable, (A2). and . (A3). f is continuous and satisfies for any

7. Regularization 2 When solving the equation numerically, we take instead of . Originally, we discretize u as for k = 1,2 …n. Here uk approximates the solution at the time tk = k , where is the time step.

8. Regularization 3 Instead, we consider the following scheme for k =1,…n with .

9. Regularization 4 • Weak form of the scheme (Problem WP) Given , find such that for all , the following equation holds

10. Error estimates 1 • Some elementary identities to be used. Here

11. Error estimates 2 Theorem 1 (apriori estimate): Assume (A1), (A2) and (A3). Then for , if solves WP, we have

12. Error estimates 3 Notation: For any is integrable in time, define Errors:

13. Error estimate 4 Theorem 2: Assume (A1)-(A3). If u is the weak solution and solves WP, then where for and k=1,…n. ( )

14. Error estimates 5 Proof : denotes the Green Operator defined by we have

15. Summary • Degenerate parabolic equation, weak solution, estimates, convergent. • Other numerical methods, similar results.

16. Thank you for attention!! Questions?