Numerical Solution of Boundary Value Problems with a Homogeneous Linear Partial Differential Equation by a Modified Bubnov-Galerkin Method on a Rectangle

   Introduction. This paper considers the numerical solution of a boundary value problem with a homogeneous linear partial differential equation of elliptic type on a rectangular domain using a modified Bubnov-Galerkin method. The unknown function is assumed to be zero at the vertices of the rectangle. Boundary value problems of this type include problems governed by the second-order Poisson and Laplace equations, the fourth-order biharmonic equation, and others. The developed algorithms allow equations with both constant and variable coefficients on the rectangular domain. The
numerical solution of the boundary value problem is represented in a functional form, i. e., as a sum of coordinate functions with an unknown vector of expansion coefficients.

   Materials and Methods. Due to the orthogonality of the proposed solution algorithms, the residual of the homogeneous linear equation is orthogonal over the rectangular domain to all coordinate functions of polynomial type, starting from the zero-index function that is identically equal to unity. All coordinate functions possess the unit Chebyshev norm on the rectangle. The inverse matrix required for solving the system of linear algebraic equations in the boundary value problem with the Poisson equation on the rectangle was computed using the Msimsl library.

   Results. The numerical solution of the boundary value problem with the Poisson equation on a rectangle using the modified Bubnov-Galerkin method shows that the uniform Chebyshev norm of the residual of the boundary value problem is of order 10−12 and is comparable with the sweep method for a pentadiagonal matrix, in which the equation itself is approximated with eighth-order accuracy using the same number of nodes of a uniform grid. However, the computation time of the modified Bubnov-Galerkin method is 30 times smaller than that of the pentadiagonal sweep method for solving the same problem. High computational efficiency is the main advantage of the Bubnov-Galerkin methods for solving boundary value problems with partial differential equations without a noticeable loss of accuracy. The optimal number of coordinate functions in the problem is fourteen.

   Discussion. The paper proposes algorithm (3)–(18) for solving the general problem with a homogeneous linear partial differential equation of arbitrary order m, and algorithm (19)–(26) for solving the Poisson (Laplace) equation on a rectangle using the modified Bubnov-Galerkin method. It should be noted that the symmetry of boundary conditions at the stage of reduction of the general problem leads to a decrease in the number of elementary subproblems. For the first time, an integral quadrature formula on a rectangle is proposed for computing the scalar product of two functions with twelfth-order accuracy.

   Conclusion. Using algorithms (28) and (29), four examples of the Poisson equation on a rectangular domain were solved numerically, where the unknown function is zero at the vertices of the rectangle. An algorithm (5)–(19) was developed for solving boundary value problems with homogeneous elliptic partial differential equations of arbitrary order, demonstrating the successful application of the modified Bubnov-Galerkin method for solving boundary value problems on a rectangular domain.

1. Bahvalov N.S., Zhidkov N.P., Kobelkov G.M. Numerical methods : a textbook for students of physics and mathematics specialties of higher educational institutions. Moscow: BINOM. Knowledge laboratory; 2011. 636 p. (In Russ.)

2. Bahvalov N.S., Lapin A.V., Chizhonkov E.V. Numerical methods in problems and exercises. Moscow: BINOM. Knowledge laboratory; 2010. 240 p. (In Russ.)

3. Pikulin V.P., Pokhozhaev S.I. Practical course on the equations of mathematical physics. Moscow: MTSNMO; 2004. 208 p. (In Russ.)

4. Sukhinov A.I., Protsenko E.A. Parallel numerical implementation of wave hydrodynamics models taking into account the features of vertical turbulent exchange. Mathematical Modeling. 2023;35(12):51–68. (In Russ.) doi: 10.20948/mm-2023-12-04

5. Sukhinov A.I., Chistyakov A.E., Nikitina A.V., Atayan A.M., Litvinov V.N. Method for solving grid equations for hydrodynamic problems in flat areas. Mathematical Modeling. 2023;35(3):35–58. (In Russ.) doi: 10.20948/mm-2023-03-03

6. Sukhinov A.I., Chistyakov A.E., Sidoryakina V.V., Kuznetsova I.I., Atayan A.M. The use of parallel computing to assess the process of transport of pollutants in shallow water bodies. News of Saratov University. New series. Series: Mathematics. Mechanics. Informatics. 2024;24(2):298–315. (In Russ.) doi: 10.18500/1816-9791-2024-24-2-298-315

7. Sidoryakina V.V., Sukhinov A.I. Construction and study of the proximity of solutions in L2 of two boundary value problems for the model of transport of multicomponent suspended matter in coastal systems. Journal of Computational Mathematics and Mathematical Physics. 2023;63(10):1721–1732. (In Russ.) doi: 10.31857/S0044466923100149

8. Sidoryakina V.V., Solomaha D.A. Symmetrized versions of the Seidel and upper relaxation methods for solving two-dimensional difference problems of elliptic. Computational Mathematics and Information Technologies. 2023;7(3):12–19. (In Russ.) doi: 10.23947/2587-8999-2023-7-3-12-19

9. Burova I.G., Altsybeev G.O. Application of second-order spline approximations to the solution of integral equations of the second kind. Computational methods and programming. 2025;26(2):175–191. (In Russ.) doi: 10.26089/NumMet.v26r213

10. Chernyshev A.D., Goryainov V.V., Kuznetsov S.V., Nikiforova O.Yu. Application of fast expansions to construct exact solutions to the problem of deflection of a rectangular membrane under the action of a variable load. Bulletin of Tomsk State University. Mathematics and mechanics. 2021;70:127–142. (In Russ.) doi: 10.17223/19988621/70/11

11. Volosova N.K., Volosov K.A., Volosova A.K., Pastukhov D.F., Pastukhov Yu.F. Modified Bubnov-Galerkin method for solving boundary value problems with a linear ordinary differential equation. Computational Mathematics and Information Technologies. 2024;8(3):23–33. (In Russ.) doi: 10.23947/2587-8999-2024-8-3-23-33

12. Volosova N.K., Volosov K. A., Volosova A. K., Pastukhov D.F., Pastukhov Yu.F. Increasing the accuracy of solving boundary value problems with linear ordinary differential equations by the Bubnov-Galerkin method. Computational Mathematics and Information Technologies. 2024;8(4):7–18. (In Russ.) doi: 10.23947/2587-8999-2024-8-4-7-18

13. Volosova N.K., Volosov K. A., Volosova A. K., Pastukhov D.F., Pastukhov Yu.F. Solving boundary value problems for some nonlinear differential equations by the Bubnov-Galerkin method. Computational Mathematics and Information Technologies. 2025; 9(1):7−19. (In Russ.) doi: 10.23947/2587-8999-2025-9-1-7-19

14. Pastukhov D.F., Pastukhov Yu.F. Approximation of the Poisson equation on a rectangle of increased accuracy. Bulletin of Polotsk State University. Series C. Fundamental sciences. 2017;12:62–77. (In Russ.)