Numerical solution of an elliptic problem with several interfaces
elliptic interface problem
discontinuity of solution
least-squares collocation method
parallelization using OpenMP
An algorithm of the high-accuracy numerical solution of the second order elliptic equation with several interfaces including intersecting and non-convex ones is developed. To approximate the interface problem in the neighbourhood of the discontinuity lines irregular cells (i-cells) which are cut off by the discontinuity lines from the regular cells of the rectangular grid and the “outsidethe-contour” parts of the cells are used. To construct an approximate solution, it is proposed: 1) to write out the additionally matching conditions in i-cells on interfaces increasing the number of matching cells; 2) to reduce the common part of the discontinuity line enclosed between neighboring cells and used for setting conditions. To solve the Dirichlet boundary value problem the hp-version of the least-squares collocation method (hp-LSCM) is implemented in combination with modern algorithms for accelerating the iterative process: preconditioning, parallelization of the computational program using OpenMP, Krylov subspaces; multigrid method. The convergence of the hp-LSCM and the conditionality of the arising overdetermined systems of linear algebraic equations (SLAE) are investigated in solving various test problems. The results obtained by the LSCM and other authors using the method MIB (matched interface and boundary) are compared.
Methods and algorithms of computational mathematics and their applications
- O. A. Oleinik, “Equations of Elliptic and Parabolic Type with Discontinuous Coefficients,” Usp. Mat. Nauk 14 (5), 164-166 (1959).
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=9165&option_lang=eng . Cited June 30, 2022.
- V. I. Isaev, A. N. Cherepanov, and V. P. Shapeev, “Numerical Study of Heat Modes of Laser Welding of Dissimilar Metals with an Intermediate Insert,” Int. J. Heat Mass Transf. 99, 711-720 (2016).
- Q. Feng, B. Han, and P. Minev, “Sixth Order Compact Finite Difference Schemes for Poisson Interface Problems with Singular Sources,” Comput. Math. Appl. 99, 2-25 (2021).
- R. C. Harris, A. H. Boschitsch, and M. O. Fenley, “Numerical Difficulties Computing Electrostatic Potentials Near Interfaces with the Poisson-Boltzmann Equation,” J. Chem. Theory Comput. 13 (8), 3945-3951 (2017).
- A. A. Samarskii, The Theory of Difference Schemes (Nauka, Moscow, 1977; Marcel Dekker, New York, 2001).
- Y. A. Sabawi, Adaptive Discontinuous Galerkin Methods for Interface Problems , PhD Thesis (University of Leicester, Leicester, 2016).
- A. Cangiani, E. H. Georgoulis, and Y. A. Sabawi, “Adaptive Discontinuous Galerkin Methods for Elliptic Interface Problems,” Math. Comp. 87, 2675-2707 (2018).
- L. Zhilin, “Fast Iterative Algorithm for Elliptic Interface Problems,” SIAM J. Numer. Anal. 35 (1), 230-254 (1998).
- C.-N. Tzou and S. N. Stechmann, “Simple Second-Order Finite Differences for Elliptic PDEs with Discontinuous Coefficients and Interfaces,” Commun. App. Math. Comp. Sci. 14 (2), 121-147 (2019).
- D. Bochkov and F. Gibou, “Solving Elliptic Interface Problems with Jump Conditions on Cartesian Grids,” J. Comput. Phys. 407 (2020).
- K. Xia, M. Zhan, and G.-W. Wei, “MIB Method for Elliptic Equations with Multi-Material Interfaces,” J. Comput. Phys. 230 (12), 4588-4615 (2011).
- Y. Chen, S. Hou, and X. Zhang, “A Bilinear Partially Penalized Immersed Finite Element Method for Elliptic Interface Problems with Multi-Domain and Triple-Junction Points,” Results Appl. Math. 8 (2020).
- V. P. Shapeev, V. A. Belyaev, and L. S. Bryndin, “High Accuracy Numerical Solution of Elliptic Equations with Discontinuous Coefficients,” Vestn. Yuzhn. Ural. Gos. Univ. Ser. Mat. Model. Programm. 14 (4), 88-101 (2021).
- V. A. Belyaev, “On the Effective Implementation and Capabilities of the Least-Squares Collocation Method for Solving Second-Order Elliptic Equations,” Vychisl. Metody Program. 22 (3), 211-229 (2021).
- B. P. Kolobov, Zh. L. Korobitsyna, A. V. Plyasunova, and A. G. Sleptsov, “A Collocation-Grid Method on Moving Grids for the Numerical Modelling of Boundary Layers,” Zh. Vychisl. Mat. Mat. Fiz. 30 (4), 521-534 (1990) [USSR Comput. Math. Math. Phys. 30 (2), 120-129 (1990)].
- V. A. Belyaev, L. S. Bryndin, S. K. Golushko, et al., “H-, P-, and HP-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications,” Zh. Vychisl. Mat. Mat. Fiz. 62 (4), 531-552 (2022) [Comput. Math. Math. Phys. 62 (4), 517-537 (2022).
- E. V. Vorozhtsov and V. P. Shapeev, “On the Efficiency of Combining Different Methods for Acceleration of Iterations at the Solution of PDEs by the Method of Collocations and Least Residuals,” Appl. Math. Comput. 363 (2019).
- M. Ramšak and L. Škerget, “A Subdomain Boundary Element Method for High-Reynolds Laminar Flow Using Stream Function-Vorticity Formulation,” Int. J. Numer. Meth. Fluids 46 (8), 815-847 (2004).
- J. M. Ortega, Introduction to Parallel and Vector Solution of Linear Systems (Plenum, New York, 1988; Mir, Moscow, 1991). doi10.1007/978-1-4899-2112-3.
- Y. Saad, Numerical Methods for Large Eigenvalue Problems (SIAM, Philadelphia, 2011).
- V. A. Belyaev, “Solving a Poisson Equation with Singularities by the Least-Squares Collocation Method,” Sib. Zh. Vychisl. Mat. 23 (3), 249-263 (2020).
doi 10.15372/SJNM20200302 [Numer. Anal. Appl. 13 (3), 207-218 (2020).
- R. P. Fedorenko, Introduction to Computational Physics (Moscow Inst. Phys. Technol., Moscow, 1994) [in Russian].
- V. I. Isaev and V. P. Shapeev, “High-Accuracy Versions of the Collocations and Least Squares Method for the Numerical Solution of the Navier-Stokes Equations,” Zh. Vychisl. Mat. Mat. Fiz. 50 (10), 1758-1770 (2010) [Comput. Math. Math. Phys. 50 (10), 1670-1681 (2010).
- V. I. Isaev, V. P. Shapeev, and S. A. Eremin, “An Investigation of the Collocation and the Least Squares Method for Solution of Boundary Value Problems for the Navier-Stokes and Poisson Equations,” Vychisl. Tekhnol. 12 (3), 53-70 (2007).
https://www.elibrary.ru/item.asp?id=12878946 . Cited June 30, 2022.
- V. P. Shapeev and A. V. Shapeev, “Solutions of the Elliptic Problems with Singularities Using Finite Difference Schemes with High Order of Approximation,” Vychisl. Tekhnol. 11 (special issue, part 2), 84-91 (2006).
https://www.elibrary.ru/item.asp?id=15281780 . Cited June 30, 2022.
- V. A. Belyaev and V. P. Shapeev, “Solving the Dirichlet Problem for the Poisson Equation by the Least Squares Collocation Method with Given Discrete Boundary Domain,” Vychisl. Tekhnol. 23 (3), 15-30 (2018).
https://elibrary.ru/item.asp?id=35095975 . Cited June 30, 2022.
- B. V. Semisalov, L. S. Bryndin, V. A. Belyaev, and A. G. Gorynin, “Numerical Analysis of Steady Polymer Fluid Flows and Its Verification,” in Proc. XXI All-Russian Conf. of Young Scientists on Mathematical Modeling and Information Technology, Novosibirsk, Russia, December 7-11, 2020 (Center Inform. Comput. Tekhnol., Novosibirsk, 2020), pp. 18-19.
https://elibrary.ru/item.asp?id=44687787 . Cited June 30, 2022.
- H. Guo, Z. Zhang, and Q. Zou, “A C^0 Linear Finite Element Method for Biharmonic Problems,” J. Sci. Comput. 74, 1397-1422 (2018).
- M. Ben-Artzi, I. Chorev, J.-P. Croisille, and D. Fishelov, “A Compact Difference Scheme for the Biharmonic Equation in Planar Irregular Domains,” SIAM J. Numer. Anal. 47 (4), 3087-3108 (2009).
- V. V. Belyaev and V. P. Shapeev, “The Collocation and Least Squares Method on Adaptive Grids in a Domain with a Curvilinear Boundary,” Vychisl. Tekhnol. 5 (4), 13-21 (2000).
https://www.elibrary.ru/item.asp?id=13026317 . Cited June 30, 2022.
- V. P. Shapeev and E. V. Vorozhtsov, “CAS Application to the Construction of the Collocations and Least Residuals Method for the Solution of the Burgers and Korteweg-de Vries-Burgers Equations,” in Lecture Notes in Computer Science (Springer, Heidelberg, 2014), Vol. 8660, pp. 432-446.