Estabilidade do método de diferenças finitas exponencial de alta ordem para problemas elípticos não lineares bidimensionais via princípio do máximo discreto

ABSTRACT

Keywords: Nonlinear elliptic equations; exponential finite difference method; discrete maximum principle; high order of convergence; interdisciplinarity.

RESUMO

Palavras-chave: Equações elípticas não lineares; método de diferenças finitas exponencial; princípio do máximo discreto; alta ordem de convergência; interdisciplinaridade.

1 INTRODUCTION

Elliptic partial differential equations (PDEs) play a fundamental role in modeling various physical, engineering, and applied science phenomena, such as steady-state heat conduction problems, electrostatics, fluid mechanics, and elasticity. In many real-world situations, these equations exhibit nonlinear characteristics over two-dimensional domains, making it impossible to obtain analytical solutions in terms of elementary functions. Consequently, the development and analysis of efficient, stable, and high-precision numerical methods become indispensable for approximating the solutions to these problems (Thomas, 1995).

Among the widely used discretization techniques, the finite difference method (FDM) (Cuminato and Junior, 2013) stands out due to its conceptual simplicity and ease of implementation. However, conventional finite difference schemes – such as the standard formulation based on a five-point stencil – exhibit an order of convergence limited to , where represents the mesh discretization step size. To achieve higher accuracy without requiring excessive mesh refinement (which would significantly increase computational cost), high-order schemes based on nine-point stencils have been developed and explored in the literature.

In this context, the EFDM approach proposed by (Pandey, 2013) stands out. This technique combines the use of compact nine-point stencils with exponential terms derived from meticulous Taylor series expansions. As a result, a significant gain in the accuracy of the Laplacian operator approximation is achieved, raising the convergence rate to .

subject to the boundary condition:

The existence and uniqueness of a solution for this class of non-linear elliptic equations, under suitable regularity conditions on and , are consolidated results in the classical theory of PDEs, see for example (Evans, 2022 and, Gilbarg and Trudinger, 1977). For the analysis and theoretical guarantee of the stability of the numerical solution obtained by the nine-point operator, use is made of the Discrete Maximum Principle and the derivation of an a priori estimate. From these theoretical results, it is demonstrated that the global error of the method is of the order of , rigorously proving the convergence and efficiency of the studied numerical scheme.

2 PROBLEM FORMULATION

In order to increase the order of convergence relative to the conventional five-point scheme, this section presents the EFDM, based on a compact nine-point stencil. This approach to the two-dimensional Dirichlet problem is inspired by the works of (Cuminato and Junior, 2013) and (Pandey, 2013).

In the following section, the local truncation error analysis of the numerical method (2.1) will be performed, demonstrating that its order of accuracy is .

3. CONSISTENCY

for some positive constant , independent of .

Adding these four equalities, we obtain

Employing once again the Taylor series expansion of the function , around the point , and truncating this expansion up to the sixth order, we have:

Summing the previous equalities, using (1.1) and the fact that

we have

From (3.3) and (3.4), we conclude that

As a consequence of (3.5), we see that

Using this with the hypotheses on the partial derivatives, we obtain (3.2), which completes the proof. Therefore, the method given in (2.1) is consistent and of order .

4. STABILITY AND CONVERGENCE

Next, we will prove the discrete maximum principle (Strikwerda, 2004).

Theorem 4.1 (Discrete Maximum Principle).

and, consequently,

Therefore, item (ii) is proven, and this completes the proof of the theorem.

The next result provides a bound for the solution to equation (2.2).

Verification of (b): It is analogous to item (a).

and

Which completes the proof.

for some positive constant , independent of .

Therefore, from (3.2) and (4.8), we conclude that the numerical method (2.2) is convergent.

5. CONCLUSION

The main results obtained in this work are highlighted below:

• High-order consistency: The Taylor series expansion with truncation at the sixth order guaranteed the fourth-order consistency of the discretization.
• Central Role of the Discrete Maximum Principle: The formal establishment of the Discrete Maximum Principle (Theorem 4.1) for the nine-point discrete operator constituted the indispensable foundation for the stability analysis of the scheme. From this principle, an a priori estimate (Theorem 4.2) was derived, bounding the norm of the discrete solution as a function of the difference operator.
• Guarantee of Stability and Convergence: By applying the a priori estimate – grounded in the Discrete Maximum Principle – to the global error, the stability and convergence of the numerical method were ensured.

It is concluded that the high-order exponential method grounded in the Discrete Maximum Principle combines theoretical soundness and computational efficiency for solving the analyzed class of problems. As future work, the extension of this approach to three-dimensional problems is suggested, as well as conducting numerical experiments for computational validation of the theoretical convergence rates.

REFERENCES

CUMINATO, J. A.; JUNIOR, M. M. Discretização de equações diferenciais parciais: técnicas de diferenças finitas. 1. ed. Rio de Janeiro: SBM, 2013.

EVANS, L. C. Partial differential equations, v. 19. American mathematical society, 2022.

GILBARG, D., TRUDINGER, N. S. Elliptic partial differential equations of second order. Vol. 224. No. 2. Berlin: springer, 1977.

PANDEY, P. K. A higher accuracy exponential finite difference method for the numerical solution of second order elliptic partial differential equations. Journal of Mathematical and Computational Science, v. 3, n. 5, p. 1325-1334, 2013.

STRIKWERDA, J. C. Finite Difference Schemes and Partial Differential Equations. 2. ed. Philadelphia: SIAM, 2004.

THOMAS, J. W. Numerical Partial Differential Equations: Finite Difference Methods. New York: Springer-Verlag, 1995.

1. Universidade Estadual de Montes Claros – Montes Claros – Minas Gerais – Brasil. ORCID: https://orcid.org/0009-0007-1159-8916 ↑

2. Escola Estadual Doutor Odilon Loures – Bocaiuva – Minas Gerais – Brasil. ORCID: https://orcid.org/0009-0009-1991-7939 ↑