The treatment of the Neumann boundary conditions for a new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes

2020

Arch Appl Mech

Self Cite

A new numerical approach for the time-dependent wave and heat equations as well as for the timeindependent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

Section: Remarkmentioning

confidence: 99%

Section: Neumann Boundary Conditions (Without the Inclusion Of Boundary Degrees Of Freedom)mentioning

confidence: 99%

Section: Neumann Boundary Conditions (With No Inclusion Of Boundary Degrees Of Freedom)mentioning

confidence: 99%

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A new numerical approach to the solution of PDEs with optimal accuracy on irregular domains and Cartesian meshes—Part 1: the derivations for the wave, heat and Poisson equations in the 1-D and 2-D cases

2020

Arch Appl Mech

Self Cite

“…These results are different from those for OLTEM developed for the scalar wave and heat equations and for the Poisson equation. In these cases OLTEM with 9‐point stencils can provide a much higher order of accuracy compared to that for linear finite elements; for example, see our papers 32,33 A special procedure has been developed for the Neumann boundary conditions on irregular boundaries.…”

Section: Discussionmentioning

confidence: 99%

“…In these cases OLTEM with 9-point stencils can provide a much higher order of accuracy compared to that for linear finite elements; for example, see our papers. 32,33 • A special procedure has been developed for the Neumann boundary conditions on irregular boundaries. It is based on adding the known Neumann boundary conditions in a number of boundary points to the right-hand side of the 9-point stencils without changing their size.…”

Section: Discussionmentioning

confidence: 99%

Optimal local truncation error method to solution of 2‐D time‐independent elasticity problems with optimal accuracy on irregular domains and unfitted Cartesian meshes

Numerical Meth Engineering

Dey

2022

Recently we have developed the optimal local truncation error method (OLTEM) for scalar PDEs on irregular domains and unfitted Cartesian meshes. Here, OLTEM is extended to a much more general case of a system of PDEs for the 2-D time-independent elasticity equations on irregular domains. Compact 9-point uniform and nonuniform stencils (with the computational costs of linear finite elements) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error.It is shown that the second order of accuracy is the maximum possible accuracy for 9-point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM is much more accurate than linear finite elements and than quadratic finite elements (up to engineering accuracy of 0.1%-1%) at the same numbers of degrees of freedom. Due to the computational efficiency and trivial unfitted Cartesian meshes that are independent of irregular domains, the proposed technique with no remeshing for the shape change of irregular domains will be effective for many engineering applications.

Optimal local truncation error method for 2‐D elastodynamics problems on irregular domains and unfitted Cartesian meshes

Num Anal Meth Geomechanics

Dey²

2022

The optimal local truncation error method (OLTEM) has been developed for 2‐D time‐dependent elasticity problems on irregular domains and trivial unfitted Cartesian meshes. Compact nine‐point uniform and nonuniform stencils (similar to those for linear finite elements on uniform meshes) are used with OLTEM. The stencil coefficients are assumed to be unknown and are calculated by the minimization of the local truncation error. It is shown that the second order of accuracy is the maximum possible accuracy for nine‐point stencils independent of the numerical technique used for their derivations. The special treatment of the Neumann boundary conditions has been developed that does not increase the size of the stencils. The cases of the nondiagonal and diagonal mass matrices are considered for OLTEM. The results of numerical examples are in agreement with the theoretical findings. They also show that due to the minimization of the local truncation error, OLTEM with the nondiagonal mass matrix is much more accurate than linear finite elements and than quadratic and cubic finite elements (up to the engineering accuracy of 0.1%–1%) at the same numbers of degrees of freedom. The proposed numerical technique can be efficiently used for many engineering applications including geomechanics.