Abstract

Finite difference schemes and iterative methods of solving anisotropic diffusion problems governing multidimensional elliptic PDE with mixed derivatives are considered. By the example of the test problem with discontinuous coefficients, it is shown that the spectral characteristics of the finite difference problem and the efficiency of their preconditioning depend on the mixed derivatives approximation method. On the basis of the comparative numerical analysis, the most adequate approximation formulas for the mixed derivatives providing a maximum convergence rate of the bi-conjugate gradients method with the incomplete LU factorization and the Fourier – Jacobi preconditioners are discovered. It is shown that the monotonicity of the finite difference scheme does not guarantee advantages at their iterative implementation. Moreover, the grid maximum principle is not provided under the conditions of essential anisotropy.

Highlights

  • Finite difference schemes and iterative methods of solving anisotropic diffusion problems governing multi­ dimensional elliptic PDE with mixed derivatives are considered

  • By the example of the test problem with discontinuous coef­ ficients, it is shown that the spectral characteristics of the finite difference problem and the efficiency of their preconditioning depend on the mixed derivatives approximation method

  • It is shown that the mo­notonicity of the finite difference scheme does not guarantee advantages at their iterative implementation

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Summary

Introduction

Finite difference schemes and iterative methods of solving anisotropic diffusion problems governing multi­ dimensional elliptic PDE with mixed derivatives are considered. П. Консервативные разностные схемы, удовлетворяющие дискретному принципу максимума (монотонности), для эллиптических уравнений со смешанными производными предложены и исследованы в работах [4,5,6], где также показано, что обеспечить выполнение этих двух характеристик не удается без дополнительных условий. В данной работе на примере модельной задачи проведен сравнительный анализ двух разностных схем с точки зрения эффективности их итерационной реализации методом би-сопряженных градиентов с различными типами переобусловливателей.

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Conclusion

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