Triangular finite differences using bivariate Lagrange polynomials with applications to elliptic equations

R. Itzá Balam, M. Uh Zapata and U. Iturrarán-Viveros

Mathematics and Computers in Simulation (MATCOM), 2025, vol. 227, issue C, 121-148

Abstract: This paper proposes finite-difference schemes based on triangular stencils to approximate partial derivatives using bivariate Lagrange polynomials. We use first-order partial derivative approximations on triangles to introduce a novel hexagonal scheme for the second-order partial derivative on any rotated parallelogram grid. Numerical analysis of the local truncation errors shows that first-order partial derivative approximations depend strongly on the triangle vertices getting at least a first-order method. On the other hand, we prove that the proposed hexagonal scheme is always second-order accurate. Simulations performed at different triangular configurations reveal that numerical errors agree with our theoretical results. Results demonstrate that the proposed method is second-order accurate for the Poisson and Helmholtz equation. Furthermore, this paper shows that the hexagonal scheme with equilateral triangles results in a fourth-order accurate method to the Laplace equation. Finally, we study two-dimensional elliptic differential equations on different triangular grids and domains.

Keywords: Finite difference; Triangular grid; Hexagonal scheme; Bivariate Lagrange polynomial; Elliptic differential equations; Koch Snowflake (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:matcom:v:227:y:2025:i:c:p:121-148

DOI: 10.1016/j.matcom.2024.07.037

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