Superconvergence of Solution Derivatives of the Shortley–Weller Difference Approximation to Elliptic Equations with Singularities Involving the Mixed Type of Boundary Conditions

Abstract

This paper presents a superconvergence analysis for the Shortley–Weller finite difference approximation of second-order self-adjoint elliptic equations with unbounded derivatives on a polygonal domain with the mixed type of boundary conditions. In this analysis, we first formulate the method as a special finite element/volume method. We then analyze the convergence of the method in a finite element framework. An O(h ^1.5)-order superconvergence of the solution derivatives in a discrete H ¹ norm is obtained. Finally, numerical experiments are provided to support the theoretical convergence rate obtained.

Keywords:

• Boundary singularity
• Finite difference method
• Mixed type of boundary conditions
• Poisson's equation
• Polygonal domains
• Shortley–Weller approximation
• Solution derivatives
• Stretching function
• Superconvergence

AMS Subject Classification:

• 65N10
• 65N30

ACKNOWLEDGMENTS

We are grateful to Professor T. Yamamoto for his valuable comments on this paper.

This work was partially supported by Scientific Research Grant-in-Aid from JSPS (No.18540107).

Notes

¹This implies that the high-order derivatives with respect to y used are all bounded. However, the derivatives with respect to x are based on (3.5).

²If the function c is piecewise highly smooth, the same superconvergence can be achieved if its discontinuity boundary is of the difference grids.

³When g _R  = O(u _n ) near the singular boundary Γ _U , we still have the bound: by following the proof of Lemma 4.8.

⁴Strictly speaking, when 3p − 2 = −1 (i.e., , the bounds in (5.10) should be modified as , and the final bound in (5.4) is also retained.