arXiv subjects

Pablo D. Brubeck

Publications and source records attributed to Pablo D. Brubeck.

11 recordsLinked to original sources

arXiv2026

Cohomology of Finite Element Stokes Complexes on Alfeld Splits

We show that the cohomology of the finite element Stokes complex consisting of piecewise polynomials spaces on an Alfeld split mesh from Fu, Guzmán, & Neilan (2020, Math. Comp., 89, 1059--1091) is isomorphic to the cohomologies of the continuous Stokes and de Rham complexes. We also construct novel "minimal" conforming finite element complexes where the $H^1$-conforming space is the lowest-order space from Guzmán & Neilan (2018, SIAM J. Numer. Anal., 56, 2826--2844) and the $L^2$-conforming space is piecewise constants. These minimal complexes also have cohomologies isomorphic to the continuous Stokes and de Rham complexes. We further construct local, bounded, cochain projections for the minimal complexes. All the results hold for strongly Lipschitz domains with nontrivial topologies and in the presence of mixed boundary conditions.

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arXiv2026

Fast solvers for the high-order FEM simplicial de Rham complex: Extended edition

We present new finite elements for solving the Riesz maps of the de Rham complex on triangular and tetrahedral meshes at high order. The finite elements discretize the same spaces as usual, but with different basis functions, so that the resulting matrices have desirable properties. These properties mean that we can solve the Riesz maps to a given accuracy in a $p$-robust number of iterations with $\mathcal{O}(p^6)$ flops in three dimensions, rather than the naïve $\mathcal{O}(p^9)$ flops. The degrees of freedom build upon an idea of Demkowicz et al., and consist of integral moments on an equilateral reference simplex with respect to a numerically computed polynomial basis that is orthogonal in two different inner products. As a result, the interior-interface and interior-interior couplings are provably weak, and we devise a preconditioning strategy by neglecting them. The combination of this approach with a space decomposition method on vertex and edge star patches allows us to efficiently solve the canonical Riesz maps at high order. We apply this to solving the Hodge Laplacians of the de Rham complex with novel augmented Lagrangian preconditioners.

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arXiv2025

DefElement: an encyclopedia of finite element definitions

DefElement is an online encyclopedia of finite element definitions that was created and is maintained by the authors of this paper. DefElement aims to make information about elements defined in the literature easily available in a standard format. There are a number of open-source finite element libraries available, and it can be difficult to check that an implementation of an element in a library matches the element's definition in the literature or implementation in another library, especially when many libraries include variants of elements whose basis functions do not match exactly. In this paper, we carefully derive conditions under which elements can be considered equivalent and describe an algorithm that uses these conditions to verify that two implementations of a finite element are indeed variants of the same element. The results of scheduled runs of our implementation of this verification algorithm are included in the information available on the DefElement website.

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arXiv2025

FIAT: enabling classical and modern macroelements

Many classical and modern finite element spaces are derived by dividing each computational cell into finer pieces. Such \emph{macroelements} frequently enable the enforcement of mathematically desirable properties such as divergence-free conditions or $C^1$ continuity in a simpler or more efficient manner than elements without the subdivision. Although a few modern software projects provide one-off support for particular macroelements, a general approach facilitating broad-based support has, until now, been lacking. In this work, we describe a major addition to the FIAT project to support a wide range of different macroelements. These enhancements have been integrated into the Firedrake code stack. We provide numerical evaluation of the new macroelement facility.

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arXiv2025

Automated Runge-Kutta-Nyström time stepping for finite element methods in Irksome

Irksome is a library based on the Unified Form Language (UFL) that automates the application of Runge-Kutta time-stepping methods for finite element spatial discretizations of partial differential equations (PDEs). This paper describes recent updates to Irksome that allow users to express semidiscrete forms of PDEs that contain second-order temporal derivatives, whence it generates stage-coupled variational problems to be solved at each time step for Runge-Kutta-Nyström methods. Firedrake then generates code for these variational problems and provides a rich interface to PETSc for solving them. Directly discretizing second-order time derivatives with Runge-Kutta-Nyström methods provides several advantages relative to discretizing a rewritten first-order system with a standard Runge-Kutta method. Besides working with an interface closer to the problem formulation in UFL, avoiding these auxiliary variables means that Runge-Kutta-Nyström methods lead to smaller algebraic systems and better run-time. Our numerical results indicate that, with effective preconditioning, fully implicit Runge-Kutta-Nyström methods can be made competitive with more traditional explicit methods for wave equations. They are also (essentially) required to discretize wave-type equations with higher-order spatial derivatives. We also provide numerical experiments for fully dynamic poroelasticity, a system of mixed temporal order, where our time-stepping and algebraic solvers perform effectively even as we approach the incompressible limit.

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arXiv2025

FIAT: improving performance and accuracy for high-order finite elements

FIAT (the FInite element Automatic Tabulator) provides a powerful Python library for the generation and evaluation of finite element basis functions on a reference element. This release paper describes recent improvements to FIAT aimed at improving its run time and the accuracy and efficiency of code generated using FIAT-provided information. In the first category, we have greatly streamlined the implementation of orthogonal polynomials out of which finite element bases are built. The second category comprises several more advances. For one, we have built an interface to the $\texttt{recursivenodes}$ package to enable more accurate Lagrange bases at high order. We have also implemented integral-type degrees of freedom for $H(\mathrm{div})$ and $H(\mathrm{curl})$ elements, which match the mathematical definitions of the elements more closely and also avoid loss of accuracy in interpolation. More fundamentally, we have included families of simplicial quadrature rules that require many fewer quadrature points than the Stroud rules previously used in FIAT. Finally, FIAT now provides support for fast diagonalization methods, which enable fast solution algorithms at very high order. In each case, we describe the new features in FIAT and illustrate some of the gains obtained through simple numerical tests.

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arXiv2024

A scalable and robust vertex-star relaxation for high-order FEM

Pavarino proved that the additive Schwarz method with vertex patches and a low-order coarse space gives a $p$-robust solver for symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate. We demonstrate the approach by solving the Poisson equation and a $H(\mathrm{div})$-conforming interior penalty discretization of linear elasticity in three dimensions at $p = 15$.

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arXiv2023

Multigrid solvers for the de Rham complex with optimal complexity in polynomial degree

The Riesz maps of the $L^2$ de Rham complex frequently arise as subproblems in the construction of fast preconditioners for more complicated problems. In this work we present multigrid solvers for high-order finite element discretizations of these Riesz maps with the same time and space complexity as sum-factorized operator application, i.e.~with optimal complexity in polynomial degree in the context of Krylov methods. The key idea of our approach is to build new finite elements for each space in the de Rham complex with orthogonality properties in both the $L^2$- and $H(\mathrm{d})$-inner products ($\mathrm{d} \in \{\mathrm{grad}, \mathrm{curl}, \mathrm{div}\})$ on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold--Falk--Winther and Hiptmair space decompositions, in the separable case. In the non-separable case, the method can be applied to an auxiliary operator that is sparse by construction. With exact Cholesky factorizations of the sparse patch problems, the application complexity is optimal but the setup costs and storage are not. We overcome this with the finer Hiptmair space decomposition and the use of incomplete Cholesky factorizations imposing the sparsity pattern arising from static condensation, which applies whether static condensation is used for the solver or not. This yields multigrid relaxations with time and space complexity that are both optimal in the polynomial degree.

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arXiv2021

Lightning Stokes solver

Gopal and Trefethen recently introduced "lightning solvers" for the 2D Laplace and Helmholtz equations, based on rational functions with poles exponentially clustered near singular corners. Making use of the Goursat representation in terms of analytic functions, we extend these methods to the biharmonic equation, specifically to 2D Stokes flow. Solutions to model problems are computed to 10-digit accuracy in less than a second of laptop time. As an illustration of the high accuracy, we resolve two or more counter-rotating Moffatt eddies near a singular corner.

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arXiv2021

Discontinuous collocation methods and gravitational self-force applications

Numerical simulations of extereme mass ratio inspirals, the mostimportant sources for the LISA detector, face several computational challenges. We present a new approach to evolving partial differential equations occurring in black hole perturbation theory and calculations of the self-force acting on point particles orbiting supermassive black holes. Such equations are distributionally sourced, and standard numerical methods, such as finite-difference or spectral methods, face difficulties associated with approximating discontinuous functions. However, in the self-force problem we typically have access to full a-priori information about the local structure of the discontinuity at the particle. Using this information, we show that high-order accuracy can be recovered by adding to the Lagrange interpolation formula a linear combination of certain jump amplitudes. We construct discontinuous spatial and temporal discretizations by operating on the corrected Lagrange formula. In a method-of-lines framework, this provides a simple and efficient method of solving time-dependent partial differential equations, without loss of accuracy near moving singularities or discontinuities. This method is well-suited for the problem of time-domain reconstruction of the metric perturbation via the Teukolsky or Regge-Wheeler-Zerilli formalisms. Parallel implementations on modern CPU and GPU architectures are discussed.

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arXiv2019

Vandermonde with Arnoldi

Vandermonde matrices are exponentially ill-conditioned, rendering the familiar "polyval(polyfit)" algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes the problem.

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