The Hodie Method and its Performance for Solving Elliptic Partial Differential Equations

Publisher Summary This chapter describes the Hodie method and its performance for solving elliptic partial differential equations. It discusses a new flexible, high-accuracy finite difference approximation to the elliptic partial differential equation. A rectangular mesh is put and at each mesh point an estimate is obtained as the solution of a finite difference equation. For simplicity of exposition, the chapter presents an assumption that the mesh is uniform with mesh spacing and this assumption is not essential to the method though it improves its efficiency in some cases. It is found that after the auxiliary points are chosen, the coefficients of the difference equation are determined to make the approximation exact on a given linear space of functions. The chapter presents a general discussion of the method's computational properties and potential applicability along with a comparative performance evaluation using the ELLPACK system. The usual difference equation for a second order problem can be derived by making the scheme exact on quadratic polynomials and it is automatically exact on cubic polynomials. It is found that the order of the discretization error is the same as the order of the truncation error.