# High Accuracy Numerical Solution of Elliptic Equations with Discontinuous Coefficients

V.P. Shapeev, V.A. Belyaev, L.S. BryndinWe develop an approach to constructing a new high-accuracy hp-version of the least-squares collocation (LSC) method for the numerical solution of boundary value problems for elliptic equations with a coefficient discontinuity on lines of different shapes in a problem solution domain. In order to approximate the equation and the conditions on the discontinuity of its coefficient, it is proposed to use the external parts and irregular cells (i-cells) of the computational grid which are cut off by the line of discontinuity from regular rectangular cells. The proposed approach allows to obtain solutions with a high order of convergence and high accuracy by grid refining and/or increasing the degree of the approximating polynomials both in the case of the Dirichlet conditions on the boundary of the domain and in the case of the presence of Neumann conditions on a large part of the boundary. Also, we consider the case of the problem with a discontinuity of the second derivatives of the desired solution in addition to the coefficient discontinuity at the corner points of the domain. We simulate the heat transfer process in the domain where particles of the medium move in a plane-parallel manner with a phase transition and heat release at the front of the discontinuity line. An effective combination of the LSC method with various methods of accelerating the iterative process is demonstrated: the acceleration algorithm based on Krylov subspaces; the operation of prolongation along the ascending branch of the V-cycle on a multigrid complex; parallelization. The results are compared with those of other authors on solving the considered problems.Full text

- Keywords
- numerical method; elliptic equations; coefficient discontinuity; conservation law; high accuracy.
- References
- 1. Samarskii A.A., Vabishchevich P.N. Vychislitel'naya teploperedacha [Computational Heat Transfer]. Moscow, Editorial URSS, 2003. (in Russian)

2. Isaev V.I., Cherepanov A.N., Shapeev V.P. Numerical Study of Heat Modes of Laser Welding of Dissimilar Metals with an Intermediate Insert. International Journal of Heat and Mass Transfer, 2016, vol. 99, pp. 711-720. DOI: 10.1016/j.ijheatmasstransfer.2016.04.019

3. Zhilin L. A Fast Iterative Algorithm for Elliptic Interface Problems. SIAM Journal on Numerical Analysis, 1998, vol. 35, no. 1, pp. 230-254. DOI: 10.1137/S0036142995291329

4. Godunov S.K., Zabrodin A.V., Ivanov M.I., Kraiko A.N., Prokopov G.P. Chislennoe reshenie mnogomernyh zadach gazovoy dinamiki [Numerical Solution of Multidimensional Problems of Gas Dynamics]. Moscow, Nauka, 1976. (in Russian)

5. Tzou C., Stechmann S.N. Simple Second-Order Finite Differences for Elliptic PDEs with Discontinuous Coefficients and Interfaces. Communications in Applied Mathematics and Computational Science, 2019, vol. 14, no. 2., pp. 121-147. DOI: 10.2140/camcos.2019.14.121

6. Belyaev V.A., Shapeev V.P. The Versions of Collocation and Least Residuals Method for Solving Problems of Mathematical Physics in the Trapezoidal Domains. Computational Technologies, 2017, vol. 22, no. 4, pp. 22-42. (in Russian)

7. Shapeev V.P., Golushko S.K., Bryndin L.S., Belyaev V.A. The Least Squares Collocation Method for the Biharmonic Equation in Irregular and Multiply-Connected Domains. Journal of Physics, 2019, vol. 1268, article ID: 012076. DOI: 10.1088/1742-6596/1268/1/012076

8. Vorozhtsov E.V., Shapeev V.P. On the Efficiency of Combining Different Methods for Acceleration of Iterations at the Solution of PDEs by the Method of Collocations and Least Residuals. Applied Mathematics and Computation, 2019, vol. 363, pp. 1-19. DOI: 10.1016/j.amc.2019.124644

9. Fedorenko R.P. Vvedenie v vychislitel'nuyu fiziku [Introduction to Сomputational Physics]. Moscow, Moscow Institute of Physics and Technology, 1994. (in Russian)

10. Saad Y. Numerical Methods for Large Eigenvalue Problems. Philadelphia, Society for Industrial and Applied Mathematics, 2011.

11. Degi D.V., Starchenko A.V. Numerical Solution of Navier-Stokes Equations on Computers with Parallel Architecture. Tomsk State University Journal of Mathematics and Mechanics, 2012, no. 2(18), pp. 88-98. (in Russian)

12. Shapeev V.P., Shapeev A.V. Solutions of the Elliptic Problems with Singularities Using Finite Difference Schemes with High Order of Approximation. Computational Technologies, 2006, vol. 11, no. 2, pp. 84-91. (in Russian)

13. Shapeev V.P. Bryndin L.S., Belyaev V.A. Solving Elliptic Equations in Polygonal Domains by the Least Squares Collocation Method. Bulletin of the South Ural State University. Series Mathematical Modelling, Programming and Computer Software, 2019, vol. 12, no. 3. pp. 140-152. (in Russian)