A Matrix-free Augmented High Order Compact Solver for Variable-Coefficient Biharmonic Problems
Abstract
We propose an augmented high-order compact finite difference method for biharmonic equations with clamped boundary conditions and variable coefficients. Standard mixed-type formulations introduce an auxiliary variable, but its boundary values are unavailable, leaving the resulting discrete systems globally coupled and difficult to solve at large scales. Our key contribution is the development of a new augmented formulation that treats these unavailable boundary values as additional unknowns, reduces the global coupling to a lower-dimensional Schur complement system, and yields decoupled second-order subproblems. The Schur complement is solved by matrix-free GMRES, while the subproblems are handled by FFT-based fast solvers. The method achieves fourth-order accuracy using compact stencils, and has O(nlog n) computational complexity, enabling the solution of the biharmonic equation with 10243 degrees of freedom within several minutes. To the best of our knowledge, this level of computational efficiency has not previously been achieved in either the literature or practice. Using energy estimates and Fourier analysis, we derive a new L2-estimate for Poisson equations with inexact Dirichlet boundary and then prove the convergence of the proposed scheme. We provide ample numerical experiments to confirm the accuracy, efficiency, and further apply the fast and accurate solver to triharmonic equations, high-wavenumber problems, Stokes flow, and plate bending problems.
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BibTeX
@misc{li2026matrixfree,
title = {{A Matrix-free Augmented High Order Compact Solver for Variable-Coefficient Biharmonic Problems}},
author = {Jin Li and Kejia Pan and Xu Qian and Li-Lian Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16478},
eprint = {2609.16478},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16478},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A Matrix-free Augmented High Order Compact Solver for Variable-Coefficient Biharmonic Problems
AU - Jin Li
AU - Kejia Pan
AU - Xu Qian
AU - Li-Lian Wang
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16478
UR - https://arxiv.org/abs/2609.16478
AB - We propose an augmented high-order compact finite difference method for biharmonic equations with clamped boundary conditions and variable coefficients. Standard mixed-type formulations introduce an auxiliary variable, but its boundary values are unavailable, leaving the resulting discrete systems globally coupled and difficult to solve at large scales. Our key contribution is the development of a new augmented formulation that treats these unavailable boundary values as additional unknowns, reduces the global coupling to a lower-dimensional Schur complement system, and yields decoupled second-order subproblems. The Schur complement is solved by matrix-free GMRES, while the subproblems are handled by FFT-based fast solvers. The method achieves fourth-order accuracy using compact stencils, and has O(nlog n) computational complexity, enabling the solution of the biharmonic equation with 1024^3 degrees of freedom within several minutes. To the best of our knowledge, this level of computational efficiency has not previously been achieved in either the literature or practice. Using energy estimates and Fourier analysis, we derive a new L^2-estimate for Poisson equations with inexact Dirichlet boundary and then prove the convergence of the proposed scheme. We provide ample numerical experiments to confirm the accuracy, efficiency, and further apply the fast and accurate solver to triharmonic equations, high-wavenumber problems, Stokes flow, and plate bending problems.
ER -