A Presentation of Sobolev Spaces by Means of the Fourier Transform

In this project, we use finite-difference to solve the Dirichlet and Poisson problem. Discrete version of the maximum principle is displayed to show the uniqueness and existence of the discrete solution. Then an example is given to interpret different convergence speed under different numerial schemes. Finally, we demonstrate that the discrete solution converges to the desired solution under appropriate conditioins. I. Difference Method for Dirichlet Problem Consider the Dirichlet problem { ∆u = 0 in Ω u = g in ∂Ω (1) where Ω is a bounded domain in Rd. The basic idea of the difference methods is to replace the given differential equation by a difference equation with step size h. With h→ 0, we try to show that the solutions of the difference equations converge to a solution of the differential equation. We introduce some notations in discretization first. Using an orthogonal grid of mesh with size h > 0 to cover Rd, we can represent every vertex point as (x, x, ..., x) = (n1h, n2h, ..., ndh) (2) with n1, n2, ..., nd ∈ Z. Let Rh denote the set of vertices and Ω̄h := Ω ∩ Rh (3)