Finite difference and finite element analysis for partial differential equations

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The starting point of this research is the following result obtained by Yamamoto (1998): The S-W finite difference solution for the boundary value problem-Δu+f(x,y,u)=0 in Ω, u=g on Γ=δΩ (1)with equal mesh size h in x and y directions yields O(h^3) accuracy near Γ and O(h^2) accuracy in other grid points, provided that u ∈ C^<3,1>(Ω^^-). This property is called "superconvergence".Through this project, we obtained the following results:(i) Superconvergence of the implicit finite difference scheme for the convection-diffusion problem u_t + div{-K(x,y)▽u + ua} = f(x,y) in Ω x (0,T).(ii) Convergence of inconsistent finite difference methods for (1) and acceleration of the numerical solution by stretching functions in the case where Ω is a square, a disk, or a sector.Some convergence theorems have been obtained.(iii) Precise error analysis for finite difference and finite element methods applied to two-point boundary value problems.By using the harmonic relation between the Green function and the discrete Green function, we obtained some interesting results on superconvergence.