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Fourth-order compact finite difference schemes for solving biharmonic equations with Dirichlet boundary conditions

Kejia Pan, Jin Li, Zhilin Li, Kang Fu

TL;DR

Addresses accurate, fourth-order solution of the biharmonic equation with Dirichlet boundaries. Proposes a genuine fourth-order compact finite difference scheme built on a coupled Poisson reformulation with $v=\Delta u$, applicable in 2D and 3D; maintains $2N^d$ unknowns and $O(h^{-2})$ conditioning. Proves and demonstrates $O(h^4)$ convergence for both $u$ and $\Delta u$, validated on smooth and oscillatory examples and with a Stokes-flow application. This approach offers high accuracy without increasing unknowns with dimension, benefiting elasticity, plate bending, and incompressible flow simulations.

Abstract

In this study, we propose a genuine fourth-order compact finite difference scheme for solving biharmonic equations with Dirichlet boundary conditions in both two and three dimensions. In the 2D case, we build upon the high-order compact (HOC) schemes for flux-type boundary conditions originally developed by Zhilin Li and Kejia Pan [SIAM J. Sci. Comput., 45 (2023), pp. A646-A674] to construct a high order compact discretization for coupled boundary conditions. When considering the 3D case, we modify carefully designed undetermined coefficient methods of Li and Pan to derive the finite difference approximations of coupled boundary conditions. The resultant FD discretization maintains the global fourth order convergence and compactness. Unlike the very popular Stephenson method, the number of unknows do not increase with dimensions. Besides, it is noteworthy that the condition number of the coefficient matrix increases at a rate of $O(h^{-2})$ in both 2D and 3D. We also validate the performance of the proposed genuine HOC methods through nontrivial examples.

Fourth-order compact finite difference schemes for solving biharmonic equations with Dirichlet boundary conditions

TL;DR

Addresses accurate, fourth-order solution of the biharmonic equation with Dirichlet boundaries. Proposes a genuine fourth-order compact finite difference scheme built on a coupled Poisson reformulation with , applicable in 2D and 3D; maintains unknowns and conditioning. Proves and demonstrates convergence for both and , validated on smooth and oscillatory examples and with a Stokes-flow application. This approach offers high accuracy without increasing unknowns with dimension, benefiting elasticity, plate bending, and incompressible flow simulations.

Abstract

In this study, we propose a genuine fourth-order compact finite difference scheme for solving biharmonic equations with Dirichlet boundary conditions in both two and three dimensions. In the 2D case, we build upon the high-order compact (HOC) schemes for flux-type boundary conditions originally developed by Zhilin Li and Kejia Pan [SIAM J. Sci. Comput., 45 (2023), pp. A646-A674] to construct a high order compact discretization for coupled boundary conditions. When considering the 3D case, we modify carefully designed undetermined coefficient methods of Li and Pan to derive the finite difference approximations of coupled boundary conditions. The resultant FD discretization maintains the global fourth order convergence and compactness. Unlike the very popular Stephenson method, the number of unknows do not increase with dimensions. Besides, it is noteworthy that the condition number of the coefficient matrix increases at a rate of in both 2D and 3D. We also validate the performance of the proposed genuine HOC methods through nontrivial examples.
Paper Structure (10 sections, 2 theorems, 44 equations, 2 figures, 8 tables)

This paper contains 10 sections, 2 theorems, 44 equations, 2 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Biharmonic equations
  4. Fourth order compact scheme for Poisson equations with a Neumann BC
  5. Fourth order compact schemes for Biharmonic problem in two dimension
  6. Fourth order compact schemes for Biharmonic problem in three dimension
  7. Numerical experiments
  8. Two dimensional examples
  9. Three dimensional examples
  10. Conclusion

Key Result

Theorem 1

Let $\left\{U_{ij}, V_{ij}\vert 0\leq i\leq N_x,0\leq j\leq N_y\right\}$ and $\left\{u_{ij},v_{ij}\vert 0\leq i\leq N_x,0\right.$$\left.\leq j\leq N_y\right\}$ be soultion of the finite difference scheme eq-4th-2d and the biharmonic problem eq-vP-eq-vB2, respectively. There is constants $C_u$, $C_v$

Figures (2)

  • Figure 1: The numerical solutions plot for Exampel \ref{['ex-o-2d']} with $k_1=25,k_2=5$ on $1024\times$1024 mesh.
  • Figure 2: The numerical solutions plot for Exampel \ref{['ex-o-2d']} with $k_1=5,k_2=50$ on $1024\times$1024 mesh.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Theorem 2
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5