10-th order of accuracy for numerical solution of 3-D elasticity equations for heterogeneous materials on unfitted Cartesian meshes

Abstract We have developed the Optimal Local Truncation Error Method (OLTEM) with 10-th order of accuracy on unfitted Cartesian meshes for a system of 3-D elasticity equations with smooth irregular interfaces. $$5\times 5\times 5=125$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>5</mml:mn> <mml:mo>×</mml:mo> <mml:mn>5</mml:mn> <mml:mo>×</mml:mo> <mml:mn>5</mml:mn> <mml:mo>=</mml:mo> <mml:mn>125</mml:mn> </mml:mrow> </mml:math> -point stencils (similar to those for quadratic finite elements) for elastic heterogeneous materials are used for OLTEM. There are no unknowns at the interface points between different materials; the structure of the global discrete equations is the same for homogeneous and heterogeneous materials. The calculation of 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 on unfitted Cartesian meshes, i.e., the increase by 7 orders in accuracy compared to quadratic finite elements on conformal meshes. A new post-processing procedure provides the 9-th order of accuracy for stresses in the 3-D case. Similar to basic computations it uses OLTEM with the 125-point stencils, the interface conditions and the elasticity equations. It was shown that the use of the elasticity equations for post-processing improves the accuracy of stresses by 6 orders compared to post-processing without the use of PDEs. At an accuracy of $$0.1\%$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0.1</mml:mn> <mml:mo>%</mml:mo> </mml:mrow> </mml:math> for stresses, OLTEM with the new post-processing procedure reduces the number of degrees of freedom by $$360 - 8000$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>360</mml:mn> <mml:mo>-</mml:mo> <mml:mn>8000</mml:mn> </mml:mrow> </mml:math> times compared to quadratic finite elements with similar stencils. OLTEM with the 125-point stencils yields even more accurate results than high-order finite elements with much wider stencils. OLTEM provides accurate numerical results for compressible and nearly incompressible materials.