Optimal solvers for fourth-order PDEs discretized on unstructured grids

TL;DR

This paper introduces the first provably optimal $\\mathcal{O}(N \log N)$ algorithms for solving linear systems from finite element discretizations of fourth-order PDEs on unstructured grids, leveraging new preconditioners and mixed-form discretizations.

Contribution

It presents novel optimal preconditioners for fourth-order PDE discretizations that reduce to solving Poisson equations, enabling efficient iterative solutions on unstructured grids.

Findings

Achieved $\mathcal{O}(N \log N)$ complexity for the linear systems.

Proven uniform convergence of conjugate gradient methods with the new preconditioners.

Demonstrated that preconditioner implementation reduces to solving Poisson equations.

Abstract

This paper provides the first provable $\mathcal{O}(N \log N)$ algorithms for the linear system arising from the direct finite element discretization of the fourth-order equation with different boundary conditions on unstructured grids of size $N$ on an arbitrary polygoanl domain. Several preconditioners are presented, and the conjugate gradient methods applied with these preconditioners are proven to converge uniformly with respect to the size of the preconditioned linear system. One main ingredient of the optimal preconditioners is a mixed-form discretization of the fourth-order problem. Such a mixed-form discretization leads to a non-desirable ---either non-optimal or non-convergent--- approximation of the original solution, but it provides optimal preconditioners for the direct finite element problem. It is further shown that the implementation of the preconditioners can be reduced to the solution of several discrete Poisson equations. Therefore, any existing optimal or nearly optimal solver, such as geometric or algebraic multigrid methods, for Poisson equations would lead to a nearly optimal solver for the discrete fourth-order system. A number of nonstandard Sobolev spaces and their discretizations defined on the boundary of polygonal domains are carefully studied and used for the analysis of those preconditioners.