Error estimates for degenerate parabolic equation Yabin Fan CASA Seminar,

Outline Introduction Weak solution Regularization Error estimates Summary

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

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.

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.

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

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

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

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

Error estimates 1 Some elementary identities to be used. Here

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

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

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

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

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

Thank you for attention!! Questions?