Compact high-order stencils with optimal accuracy for numerical solutions of 2-D time-independent elasticity equations

Abstract

A new approach for the increase in the order of accuracy of high-order numerical techniques used for time-independent elasticity is suggested on uniform square and rectangular meshes. It is based on the optimization of the coefficients of the corresponding discrete stencil equations with respect to the local truncation error. It is shown that the second order of accuracy of 2-D linear finite elements with 9-point stencils is optimal and cannot be improved. However, the order of accuracy of 25-point stencils (similar to those for quadratic finite and isogeometric elements) can be significantly improved. We have developed new 25-point stencils for 2-D elastic problems with optimal 10th order of accuracy. The numerical results are in good agreement with the theoretical findings as well as they show a big increase in accuracy of the new stencils compared with those for high-order finite elements. At the same number of degrees of freedom, the new approach yields significantly more accurate results than those obtained by high-order (up to the tenth order) finite elements. The numerical experiments also show that the new approach with 25-point stencils yields very accurate results for nearly incompressible materials with Poisson’s ratio 0.4999.