arXiv subjects

Robert C. Kirby

Publications and source records attributed to Robert C. Kirby.

At least 19 recordsLinked to original sources

arXiv2026

Bounds-Constrained Finite Element Approximation of Time-Dependent Partial Differential Equations

Finite elements provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not, in general, guarantee enforcement of bounds constraints inherent in the original problem. We propose two approaches to enforcing bounds constraints for time-dependent problems. First, we propose general projective methods which result from a systematic modification of any abstract time-stepping scheme. Second, we present a monolithic technique for which we take a modified formulation of implicit single-stage Runge-Kutta methods and general implicit multistep methods as prototypical examples. By solving a constrained optimization problem, we are able to ensure that the bounds constraints are enforced by the approximate solution at the discrete time levels, obtaining (formally) high order methods both in space and time. Numerical examples for the linear heat and advection equations and nonlinear Allen-Cahn equation are given.

math.NA↗

arXiv2026

Automated Galerkin time stepping in Irksome

As the study of temporal and spatial discretization schemes continues to advance, recent work has focused on the use of Galerkin-in-time discretization schemes that enable broader structure-preservation than is known for Runge-Kutta integrators. While the promise of such discretizations is immense, their realization has, until now, generally relied on bespoke implementations that have limited their wider use. In this work, we present automation in Irksome for both discontinuous Galerkin and continuous Petrov-Galerkin time stepping of semidiscrete variational problems. The implementation supports auxiliary variables, flexible temporal quadrature, and monolithic algebraic solvers, and it enables switching between Runge-Kutta and Galerkin-in-time formulations with minimal changes to user code. Numerical examples illustrate accuracy, solver performance, and structure preservation across representative PDE systems.

math.NA↗

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.

math.NA↗

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.

math.NA↗

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.

math.NA↗

arXiv2025

Bounds-constrained finite element approximation of time-dependent partial differential equations

Finite element methods provide accurate and efficient methods for the numerical solution of partial differential equations by means of restricting variational problems to finite-dimensional approximating spaces. However, they do not guarantee enforcement of bounds constraints inherent in the original problem. Previous work enforces these bounds constraints by replacing the variational equations with variational inequalities. We extend this approach to collocation-type Runge-Kutta methods for time-dependent problems, obtaining (formally) high order methods in both space and time. By using a novel reformulation of the collocation scheme, we can guarantee that the bounds constraints hold uniformly in time. Numerical examples for a model of phytoplankton growth, the heat equation, and the Cahn-Hilliard system are given.

math.NA↗

arXiv2025

Extending Irksome: improvements in automated Runge--Kutta time stepping for finite element methods

Irksome is a library based on the Unified Form Language (UFL) that enables automated generation of Runge--Kutta methods for time-stepping finite element spatial discretizations of partial differential equations (PDE). Allowing users to express semidiscrete forms of PDE, it generates UFL representations for the stage-coupled variational problems to be solved at each time step. The Firedrake package then generates efficient code for evaluating these variational problems and allows users a wide range of options to deploy efficient algebraic solvers in PETSc. In this paper, we describe several recent advances in Irksome. These include alternate formulations of the Runge--Kutta time-stepping methods and optimized support for diagonally implicit (DIRK) methods. Additionally, we present new and improved tools for building preconditioners for the resulting linear and linearized systems, demonstrating that these can lead to efficient approaches for solving fully implicit Runge-Kutta discretizations. The new features are demonstrated through a sequence of computational examples demonstrating the high-level interface and obtained solver performance.

math.NA↗

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.

math.NA↗

arXiv2024

High-order bounds-satisfying approximation of partial differential equations via finite element variational inequalities

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the $W^{1,p}$ norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection-diffusion problems, all subject to bounds constraints.

math.NA↗

arXiv2023

Planar Curve Registration using Bayesian Inversion

We study parameterisation-independent closed planar curve matching as a Bayesian inverse problem. The motion of the curve is modelled via a curve on the diffeomorphism group acting on the ambient space, leading to a large deformation diffeomorphic metric mapping (LDDMM) functional penalising the kinetic energy of the deformation. We solve Hamilton's equations for the curve matching problem using the Wu-Xu element [S. Wu, J. Xu, Nonconforming finite element spaces for $2m^\text{th}$ order partial differential equations on $\mathbb{R}^n$ simplicial grids when $m=n+1$, Mathematics of Computation 88 (316) (2019) 531-551] which provides mesh-independent Lipschitz constants for the forward motion of the curve, and solve the inverse problem for the momentum using Bayesian inversion. Since this element is not affine-equivalent we provide a pullback theory which expedites the implementation and efficiency of the forward map. We adopt ensemble Kalman inversion using a negative Sobolev norm mismatch penalty to measure the discrepancy between the target and the ensemble mean shape. We provide several numerical examples to validate the approach.

cs.CV↗

arXiv2023

On the convergence of monolithic multigrid for implicit Runge-Kutta time stepping of finite element problems

Finite element discretization of time dependent problems also require effective time-stepping schemes. While implicit Runge-Kutta methods provide favorable accuracy and stability problems, they give rise to large and complicated systems of equations to solve for each time step. These algebraic systems couple all Runge-Kutta stages together, giving a much larger system than for single-stage methods. We consider an approach to these systems based on monolithic smoothing. If stage-coupled smoothers possess a certain kind of structure, then the question of convergence of a two-grid or multi-grid iteration reduces to convergence of a related strategy for a single-stage system with a complex-valued time step. In addition to providing a general theoretical approach to the convergence of monolithic multigrid methods, several numerical examples are given to illustrate the theory show how higher-order Runge-Kutta methods can be made effective in practice.

math.NA↗

arXiv2023

Integral Equation Methods for the Morse-Ingard Equations

We present two (a decoupled and a coupled) integral-equation-based methods for the Morse-Ingard equations subject to Neumann boundary conditions on the exterior domain. Both methods are based on second-kind integral equation (SKIE) formulations. The coupled method is well-conditioned and can achieve high accuracy. The decoupled method has lower computational cost and more flexibility in dealing with the boundary layer; however, it is prone to the ill-conditioning of the decoupling transform and cannot achieve as high accuracy as the coupled method. We show numerical examples using a Nyström method based on quadrature-by-expansion (QBX) with fast-multipole acceleration. We demonstrate the accuracy and efficiency of the solvers in both two and three dimensions with complex geometry.

math.NA↗

arXiv2022

Exact domain truncation for the Morse-Ingard equations

Morse and Ingard give a coupled system of time-harmonic equations for the temperature and pressure of an excited gas. These equations form a critical aspect of modeling trace gas sensors. Like other wave propagation problems, the computational problem must be closed with suitable far-field boundary conditions. Working in a scattered-field formulation, we adapt a nonlocal boundary condition proposed earlier for the Helmholtz equation to this coupled system. This boundary condition uses a Green's formula for the true solution on the boundary, giving rise to a nonlocal perturbation of standard transmission boundary conditions. However, the boundary condition is exact and so Galerkin discretization of the resulting problem converges to the restriction of the exact solution to the computational domain. Numerical results demonstrate that accuracy can be obtained on relatively coarse meshes on small computational domains, and the resulting algebraic systems may be solved by GMRES using the local part of the operator as an effective preconditioner.

math.NA↗

arXiv2022

Additive Schwarz methods for serendipity elements

While solving Partial Differential Equations (PDEs) with finite element methods (FEM), serendipity elements allow us to obtain the same order of accuracy as rectangular tensor-product elements with many fewer degrees of freedom (DOFs). To realize the possible computational savings, we develop some additive Schwarz methods (ASM) based on solving local patch problems. Adapting arguments from Pavarino for the tensor-product case, we prove that patch smoothers give conditioning estimates independent of the polynomial degree for a model problem. We also combine this with a low-order global operator to give an optimal two-grid method, with conditioning estimates independent of the mesh size and polynomial degree. The theory holds for serendipity elements in two and three dimensions, and can be extended to full multigrid algorithms. Numerical experiments using Firedrake and PETSc confirm this theory and demonstrate efficiency relative to standard elements.

math.NA↗

arXiv2022

Weighted-norm preconditioners for a multi-layer tide model

We derive a linearized rotating shallow water system modeling tides, which can be discretized by mixed finite elements. Unlike previous models, this model allows for multiple layers stratified by density. Like the single-layer case~\cite{kirby2021preconditioning} a weighted-norm preconditioner gives a (nearly) parameter-robust method for solving the resulting linear system at each time step, but the all-to-all coupling between the layers in the model poses a significant challenge to efficiency. Neglecting the inter-layer coupling gives a preconditioner that degrades rapidly as the number of layers increases. By a careful analysis of the matrix that couples the layers, we derive a robust method that requires solving a reformulated system that only involves coupling between adjacent layers. Numerical results obtained using Firedrake confirm the theory.

math.NA↗

arXiv2021

Bounds-constrained polynomial approximation using the Bernstein basis

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds, which implies such bounds on the approximant. We frame the problem as an inequality-constrained optimization problem and give an algorithm for finding the Bernstein coefficients of the exact solution. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. It also extends naturally to multivariate polynomials over a simplex.

math.NA↗

arXiv2021

Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake

We present an implementation of the trimmed serendipity finite element family, using the open source finite element package Firedrake. The new elements can be used seamlessly within the software suite for problems requiring $H^1$, \hcurl, or \hdiv-conforming elements on meshes of squares or cubes. To test how well trimmed serendipity elements perform in comparison to traditional tensor product elements, we perform a sequence of numerical experiments including the primal Poisson, mixed Poisson, and Maxwell cavity eigenvalue problems. Overall, we find that the trimmed serendipity elements converge, as expected, at the same rate as the respective tensor product elements while being able to offer significant savings in the time or memory required to solve certain problems.

math.NA↗

arXiv2021

Full waveform inversion using triangular waveform adapted meshes

In this article, continuous Galerkin finite elements are applied to perform full waveform inversion (FWI) for seismic velocity model building. A time-domain FWI approach is detailed that uses meshes composed of variably sized triangular elements to discretize the domain. To resolve both the forward and adjoint-state equations, and to calculate a mesh-independent gradient associated with the FWI process, a fully-explicit, variable higher-order (up to degree $k=5$ in $2$D and $k=3$ in 3D) mass lumping method is used. By adapting the triangular elements to the expected peak source frequency and properties of the wavefield (e.g., local P-wavespeed) and by leveraging higher-order basis functions, the number of degrees-of-freedom necessary to discretize the domain can be reduced. Results from wave simulations and FWIs in both $2$D and 3D highlight our developments and demonstrate the benefits and challenges with using triangular meshes adapted to the material proprieties. Software developments are implemented an open source code built on top of Firedrake, a high-level Python package for the automated solution of partial differential equations using the finite element method.

cs.CE↗