arXiv subjects

L. Ridgway Scott

Publications and source records attributed to L. Ridgway Scott.

At least 19 recordsLinked to original sources

arXiv2026

Assessment of Numerical Lift Coefficient Data for a Circular Cylinder with Application to Bladeless Turbines

We assess computed lift coefficient data for flow past a circular cylinder to evaluate their suitability for practical applications. Specifically, we consider lift coefficient data for a circular cylinder over Reynolds numbers from 120 to 8000. The results are obtained from two-dimensional finite element simulations of the incompressible Navier-Stokes equations using pressure robust discretizations. We compare the computed lift coefficients with published experimental and numerical results, finding good agreement in some cases but significant disagreement in others. Because lift fluctuations are central to vortex-induced vibration concepts, these data therefore provide input for the analysis and preliminary design of bladeless turbines.

physics.flu-dyn↗

arXiv2026

Sufficient conditions for strong discrete maximum principles in finite element solutions of linear and semilinear elliptic equations

We introduce a novel technique for proving global strong discrete maximum principles for finite element discretizations of linear and semilinear elliptic equations for cases when the common, matrix-based sufficient conditions are not satisfied. The basic argument consists of extending the strong form of discrete maximum principle from macroelements to the entire domain via a connectivity argument. The method is applied to discretizations of elliptic equations with certain pathological meshes, and to semilinear elliptic equations.

math.NA↗

arXiv2026

Scott-Vogelius element and iterated penalty method for inhomogeneous Dirichlet boundary conditions

We present quasi-optimal a priori error estimates for general mixed finite element methods to approximate solutions of the Stokes problem subject to inhomogeneous Dirichlet boundary conditions. For the Scott--Vogelius element this yields pressure-robust a priori error stimates. Due to the exact divergence constraint, this requires a compatibility condition for the boundary data to hold. A key tool is a modified Fortin operator, capable of preserving this compatibility condition. Furthermore, we analyse the iterated penalty method, an Uzawa-type algorithm and we show its convergence and asymptotic pressure robustness. Numerical experiments support the theory and highlight the importance of the compatibility condition and the appropriate treatment of nearly singular vertices.

math.NA↗

arXiv2026

A local Fortin projection for the Scott-Vogelius elements on general meshes

We construct a local Fortin projection for the Scott-Vogelius finite element pair for polynomial degree $k \ge 4$ on general shape-regular triangulations in two dimensions. In particular, the triangulation may contain singular vertices. In addition to preserving the divergence in the dual of the pressure space, the projection preserves discrete boundary data and satisfies local stability estimates.

math.NA↗

arXiv2026

On the convergence of iterated penalty methods for structure-preserving discretizations of saddle point problems

We present new convergence estimates for the iterated penalty method applied to structure-preserving discretizations of linear generalized saddle point systems. The method may be viewed as an Uzawa iteration on an augmented Lagrangian formulation of the system. As a by-product, we obtain sharper stability estimates for penalized/perturbed saddle point problems. Three model finite element applications show agreement with the theory.

math.NA↗

arXiv2025

A Mathematical Primer on Water Ice

Water adopts many different crystal structures in its solid form. These provide insight into potential structures of water even in its liquid phase, and they can be used to calibrate pair potentials used for simulation of water. In crowded biological environments, water may behave more like ice than bulk water. The different ice structures have different dielectric properties. This brief primer is intended to facilitate further research.

physics.chem-ph↗

arXiv2025

Kinetic energy instability of pipe flow in finite domains

The instability of pipe flow has been a subject of extensive research, yet a significant gap remains between experimental observations and theoretical predictions. This study revisits the classical problem of kinetic energy instability of pipe flow using contemporary computational methods. We focus on finite domains to address the limitations of previous infinite pipe analyses. We analyze two different cases by imposing homogeneous Dirichlet and periodic boundary conditions. Our investigation reveals that the critical Reynolds number for instability approaches values established by Joseph and Carmi $(1969)$ as the length of the pipe increases. Specifically, when homogeneous Dirichlet boundary conditions are applied, the critical Reynolds number converges monotonically to a value of $\RE_c \leq 81.62$. In contrast, for the periodic case, the critical Reynolds number varies periodically with the length of the pipe, exhibiting local minima at $\RE_c = 81.58$. We further characterize the shape of the perturbations by computing their magnitude and vorticity for several local minima and maxima associated with the periodic problem. Focusing on the perturbations obtained by imposing the homogeneous Dirichlet boundary conditions, we compute their temporal evolution. As expected, the $L^2$-norm of the perturbation decreases from the initial time when solving for the critical Reynolds number, while it initially increases and then eventually decreases for Reynolds numbers exceeding the critical value.

physics.flu-dyn↗

arXiv2025

Benchmark stress tests for flow past a cylinder at higher Reynolds numbers using EMAC

We consider a test problem for Navier-Stokes solvers based on the flow around a cylinder at Reynolds numbers 500 and 1000, where the solution is observed to be periodic when the problem is sufficiently resolved. Computing the resulting flow is a challenge, even for exactly divergence-free discretization methods, when the scheme does not include sufficient numerical dissipation. We examine the performance of the energy, momentum and angular momentum conserving (EMAC) formulation of the Navier-Stokes equations. This incorporates more physical conservation into the finite element method even when the numerical solution is not exactly divergence-free. Consequently, it has a chance to outperform standard methods, especially for long-time simulations. We find that for lowest-order Taylor-Hood elements, EMAC outperforms the standard convective formulations. However, for higher-order elements, EMAC can become unstable on under-resolved meshes.

math.NA↗

arXiv2025

A Finite Element Implementation of the SRTD Algorithm for an Oldroyd 3-Parameter Viscoelastic Fluid Model

In this paper, we discuss a finite element implementation of the SRTD algorithm described by Girault and Scott for the steady-state case of a certain 3-parameter subset of the Oldroyd models. We compare it to the well-known EVSS method, which, though originally described for the upper-convected Maxwell model, can easily accommodate the Oldroyd 3-parameter model. We obtain numerical results for both methods on two benchmark problems: the lid-driven cavity problem and the journal-bearing, or eccentric rotating cylinders, problem. We find that the resulting finite element implementation of SRTD is stable with respect to mesh refinement and is generally faster than EVSS, though is not capable of reaching as high a Weissenberg number as EVSS.

math.NA↗

arXiv2024

Reliable chaotic transition in incompressible fluid simulations

We consider a test problem for Navier-Stokes solvers based on the flow around a cylinder that exhibits chaotic behavior, to examine the behavior of various numerical methods. We choose a range of Reynolds numbers for which the flow is time-dependent but can be characterized as essentially two-dimensional. The problem requires accurate resolution of chaotic dynamics over a long time interval. It also requires the use of a relatively large computational domain, part of which is curved. We review the performance of different finite element methods for the proposed range of Reynolds numbers. These tests indicate that some of the most established methods do not capture the correct behavior. The key requirements identified are pressure-robustness of the method, high resolution, and appropriate numerical dissipation when the smallest scales are under-resolved.

math.NA↗

arXiv2024

Dimensions of exactly divergence-free finite element spaces in 3D

We examine the dimensions of various inf-sup stable mixed finite element spaces on tetrahedral meshes in 3D with exact divergence constraints. More precisely, we compare the standard Scott-Vogelius elements of higher polynomial degree and low order methods on split meshes, the Alfeld and the Worsey-Farin split. The main tool is a counting strategy to express the degrees of freedom for given polynomial degree and given split in terms of few mesh quantities, for which bounds and asymptotic behavior under mesh refinement is investigated. Furthermore, this is used to obtain insights on potential precursor spaces in full de Rham complexes for finite element methods on the Worsey-Farin split.

math.NA↗

arXiv2024

Computational analysis of a contraction rheometer for the grade-two fluid model

We explore the possibility of simulating the grade-two fluid model in a geometry related to a contraction rheometer, and we provide details on several key aspects of the computation. We show how the results can be used to determine the viscosity $ν$ from experimental data. We also explore the identifiability of the grade-two parameters $α_1$ and $α_2$ from experimental data. In particular, as the flow rate varies, force data appears to be nearly the same for certain distinct pairs of values $α_1$ and $α_2$; however we determine a regime for $α_1$ and $α_2$ for which the parameters may be identifiable with a contraction rheometer.

math.NA↗

arXiv2024

Two conjectures on the Stokes complex in three dimensions on Freudenthal meshes

In recent years a great deal of attention has been paid to discretizations of the incompressible Stokes equations that exactly preserve the incompressibility constraint. These are of substantial interest because these discretizations are pressure-robust, i.e. the error estimates for the velocity do not depend on the error in the pressure. Similar considerations arise in nearly incompressible linear elastic solids. Conforming discretizations with this property are now well understood in two dimensions, but remain poorly understood in three dimensions. In this work we state two conjectures on this subject. The first is that the Scott-Vogelius element pair is inf-sup stable on uniform meshes for velocity degree $k \ge 4$; the best result available in the literature is for $k \ge 6$. The second is that there exists a stable space decomposition of the kernel of the divergence for $k \ge 5$. We present numerical evidence supporting our conjectures.

math.NA↗

arXiv2023

Chaotic dynamics of two-dimensional flows around a cylinder

We study flow around a cylinder from a dynamics perspective, using drag and lift as indicators. We observe that the mean drag coefficient bifurcates from the steady case when the Karman vortex street emerges. We also find a jump in the dimension of the drag/lift attractor just above Reynolds number 100. We compare the simulated drag values with experimental data obtained over the last hundred years. Our simulations suggest that a vibrational resonance in the cylinder would be unlikely for Reynolds numbers greater than 1000, where the drag/lift behavior is fully chaotic.

math.NA↗

arXiv2023

Verification and Validation of Cylinder Drag: Pressure and Stress Approximations on Curved Boundaries

We study a technique for verification of stress and pressure computations on boundaries in flow simulations. We utilize existing experiments to provide validation of the simulations. We show that this approach can reveal critical flaws in simulation algorithms. Using the successful computational algorithms, we examine Lamb's model for cylinder drag at low Reynolds numbers. We comment on a discrepancy observed in an experimental paper, suggesting that the domain size may be a contributing factor. Our simulations on suitably large domains confirm Lamb's model. We highlight a paradox related to imposing Dirichlet (Stokes) boundary conditions on polygonal approximations of the curved surface using finite-element methods that are exactly divergence free. The finite-element simulations provide very poor representations of drag when the boundary conditions are imposed strongly. We demonstrate that relaxing the boundary conditions using Nitsche's method restores high-order approximation.

physics.flu-dyn↗

arXiv2022

Resolution of d'Alembert's paradox using slip boundary conditions: The effect of the friction parameter on the drag coefficient

d'Alembert's paradox is the contradictory observation that for incompressible and inviscid (potential) fluid flow, there is no drag force experienced by a body moving with constant velocity relative to the fluid. This paradox can be straightforwardly resolved by considering Navier's slip boundary condition. Potential flow around a cylinder then solves the Navier--Stokes equations using friction parameter $β=-2ν$. This negative friction parameter can be interpreted physically as the fluid being accelerated by the cylinder wall. This explains the lack of drag. In this paper, we introduce the Navier slip boundary condition and show that choosing the friction parameter positive resolves d'Alembert's paradox. We then further examine the effect of the friction parameter $β$ on the drag coefficient. In particular, we show that for large $β$ the drag coefficient corresponds well with experimental values. Moreover, we provide numerical evidence that the Newton continuation method (moving from small to large Reynold's numbers) requires fewer iterations to succeed. Thus the slip boundary condition is advantageous also from a computational perspective.

physics.flu-dyn↗

arXiv2021

Robust multigrid methods for nearly incompressible elasticity using macro elements

We present a mesh-independent and parameter-robust multigrid solver for the Scott-Vogelius discretisation of the nearly incompressible linear elasticity equations on meshes with a macro element structure. The discretisation achieves exact representation of the limiting divergence constraint at moderate polynomial degree. Both the relaxation and multigrid transfer operators exploit the macro structure for robustness and efficiency. For the relaxation, we use the existence of local Fortin operators on each macro cell to construct a local space decomposition with parameter-robust convergence. For the transfer, we construct a robust prolongation operator by performing small local solves over each coarse macro cell. The necessity of both components of the algorithm is confirmed by numerical experiments.

math.NA↗

arXiv2021

A Reynolds-robust preconditioner for the Scott-Vogelius discretization of the stationary incompressible Navier-Stokes equations

Augmented Lagrangian preconditioners have successfully yielded Reynolds-robust preconditioners for the stationary incompressible Navier-Stokes equations, but only for specific discretizations. The discretizations for which these preconditioners have been designed possess error estimates which depend on the Reynolds number, with the discretization error deteriorating as the Reynolds number is increased. In this paper we present an augmented Lagrangian preconditioner for the Scott-Vogelius discretization on barycentrically-refined meshes. This achieves both Reynolds-robust performance and Reynolds-robust error estimates. A key consideration is the design of a suitable space decomposition that captures the kernel of the grad-div term added to control the Schur complement; the same barycentric refinement that guarantees inf-sup stability also provides a local decomposition of the kernel of the divergence. The robustness of the scheme is confirmed by numerical experiments in two and three dimensions.

math.NA↗