The high-order finite element Duffy de Rham complex and low-order-refined preconditioning
This work addresses computational efficiency in numerical simulations for partial differential equations, particularly in finite element methods, by enabling effective preconditioning for high-order spaces, though it is incremental as it builds on existing low-order-refined techniques.
The authors tackled the challenge of constructing high-order finite element spaces for the L² de Rham complex on triangular meshes that are amenable to efficient low-order-refined preconditioning, achieving spectral equivalence independent of polynomial degree and confirming results with numerical tests on diverse high-order problems.
In this work, we construct high-order finite element spaces for the $L^2$ de Rham complex on triangular meshes amenable to low-order-refined preconditioning. The spaces are constructed using the Duffy transformation, by pulling back appropriately chosen polynomial spaces defined on the unit square; in addition to piecewise polynomials, these spaces also contain certain rational functions, and they reduce to the standard Lagrange, Nédélec, and discontinuous finite elements in the lowest-order case. We establish spectral equivalence, independent of the polynomial degree, of the stiffness matrices defined on these spaces with the lowest-order stiffness matrices defined on refined meshes, constructed using a Gauss-Lobatto triangular lattice. Spectral equivalence of the operators is a consequence of norm equivalences in Jacobi-weighted $L^2$ norms, which are established by proving stability of the Jacobi-Gauss-Lobatto interpolation operator in shifted norms. The low-order-refined preconditioners can also be used to precondition the standard piecewise polynomial finite element spaces using a fictitious space approach. The low-order-refined system can in turn be preconditioned effectively using algebraic multigrid methods. The analytical estimates are confirmed by numerical results on a variety of high-order problems, including on mixed meshes and surface meshes.