The 10-th order of accuracy of ‘quadratic’ elements for elastic heterogeneous materials with smooth interfaces and unfitted Cartesian meshes

Recently, we have developed the optimal local truncation error method (OLTEM) for PDEs with homogeneous materials on regular and irregular domains and Cartesian meshes as well as OLTEM with simple 9-point stencils for the 2-D scalar time-dependent wave and heat equations for heterogeneous materials with irregular interfaces. Here, OLTEM is extended to a much more general case of a system of elastic PDEs for heterogeneous materials with smooth irregular interfaces and unfitted Cartesian meshes. We also use larger 25-point stencils that are similar to those for quadratic quadrilateral finite elements. The interface conditions on the interfaces where the jumps in material properties occur are added to the expression for the local truncation error and do not change the width of the stencils. There are no unknowns on interfaces between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal 10-th order of accuracy for OLTEM with the 25-point stencils on unfitted Cartesian meshes. This corresponds to the increase in accuracy by seven orders for OLTEM compared to conventional quadratic finite elements with similar stencils. A new post-processing procedure for the stress calculations has been developed. Similar to basic computations, it includes OLTEM with the 25-point compact stencils and provides a very high accuracy of the stresses. Numerical experiments for elastic heterogeneous materials with circular and elliptical interfaces show that at the same number of degrees of freedom, OLTEM with unfitted meshes is more accurate than high order (up to the fifth order—the maximum order implemented in the COMSOL software) finite elements with a much greater stencil width and conformed meshes. Moreover, OLTEM with the 25-point stencils provides very accurate results for nearly incompressible materials (e.g., with Poisson ratio 0.4995).