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Recently the optimal local truncation error method (OLTEM) has been developed for the 2-D Poisson equation for heterogeneous materials with irregular interfaces and unfitted Cartesian meshes. Here we extend it to the general 3-D case. 27-point stencils that are similar to those for linear finite elements are used with OLTEM. The interface conditions at a small number of selected interface points 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, includes the entire PDE for the derivations and yields the optimal third order of accuracy of OLTEM with the 27-point stencils. The 3-D numerical results for heterogeneous materials with irregular interfaces and different material contrasts show that at the same number of degrees of freedom, OLTEM is even much more accurate than high-order (up to the 6th order) finite elements with much wider stencils. Compared to linear finite elements with similar 27-point stencils, at the engineering accuracy of 0.1 % OLTEM decreases the number of degrees of freedom by a factor of greater than 3500. This leads to a huge reduction in computation time. For the first time, a new post-processing procedure has been developed with OLTEM for the calculation of the spatial derivatives of numerical solutions. The spatial derivatives for each grid point are calculated with the help of one compact 27-point stencil (the same as for basic computations) for the corresponding grid point and the use of the original PDE. The spatial derivatives of the OLTEM solutions calculated with the new post-processing procedure are much more accurate compared to those obtained by high-order (up to the 7th order) finite elements with much wider stencils. At the engineering accuracy of 0.1 % for the spatial derivatives, OLTEM decreases the number of degrees of freedom by a factor of greater than 10 6 compared to linear finite elements. The new post-processing procedure can be equally applied to the calculation of the partial derivatives obtained by other numerical methods as well as to the numerical results for other PDEs. Due to the huge reduction in the computation time compared to existing methods and the use of trivial unfitted Cartesian meshes that are independent of irregular geometry, the proposed technique does not require remeshing for the shape change of irregular geometry and it will be effective for many design and optimization problems as well as for multiscale problems without the scale separation.
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