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Ridgway Scott

Bio: Ridgway Scott is an academic researcher from University of Michigan. The author has contributed to research in topics: Finite element method & Galerkin method. The author has an hindex of 3, co-authored 3 publications receiving 498 citations.

Papers
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Journal ArticleDOI
TL;DR: In this paper, a polynomial plus a remainder is represented as a Taylor series and the remainder can be manipulated in many ways to give different types of bounds, including integer order and nonstandard Sobolev-like spaces.
Abstract: Constructive proofs and several generalizations of approximation results of J. H. Bramble and S. R. Hilbert are presented. Using an averaged Taylor series, we represent a function as a polynomial plus a remainder. The remainder can be manipulated in many ways to give different types of bounds. Approximation of functions in fractional order Sobolev spaces is treated as well as the usual integer order spaces and several nonstandard Sobolev-like spaces.

447 citations

Journal ArticleDOI
01 Jan 1979
TL;DR: In this article, families of C piecewise polynomial spaces of degree r + 3 on triangles and quadrilatéral in two dimensions are constructed, and approximation properties of the families are studied.
Abstract: — Families of C piecewise polynomial spaces of degree r ^ 3 on triangles and quadrilatéral in two dimensions are constructed, and approximation properties of the families are studied. Examples of the use of the families in Galerkin methodsfor 2nd and 4th order elliptic boundary value problems on arbitrarily shaped domains are given. The approximation properties on the boundary are such that the rate of convergence of the Galerkin methods is the optimal rate determined by the degree r of the piecewise polynomial space. Résumé. — En dimension deux, on construit des familles d'espaces de classe C .formés de polynômes de degré r ^ 3 par morceaux, sur des triangles et des quadrilatères, et on étudie les propriétés d'approximation de ces familles. On en donne des exemples d'application à des méthodes de Galerkin pour les problèmes aux limites elliptiques du 2 et du 4 ordre posés sur des domaines déforme arbitraire. Les propriétés d'approximation de la frontière sont telles que le taux de convergence des méthodes de Galerkin est le taux optimal, déterminé par le degré r de V'espace des polynômes par morceaux.

89 citations

Journal ArticleDOI
TL;DR: The Chow-Yorke algorithm is a homotopy method that has been proved globally convergent for a wide range of practical engineering problems, such as zero finding and nonlinear programming problems.
Abstract: The Chow-Yorke algorithm, i.e., a homotopy method that has been proved globally convergent j problems, certain classes of zero finding and nonlinear programming problems, and two-point, boundary based on shooting, finite differences, and spline collocation. The method is numerically stable and has been applied to a wide range of practical engineering problems. Here the Chow-Yorke algorithm is proved globally c Galerkin approximations to nonlinear two-point boundary value problems. Several numerical implement are briefly described, and computational results are presented for a fairly difficult, magnet-hydrodynamic problem. Key words. homotopy method, Chow-Yorke algorithm, globally convergent,, two-point boundary value method, finite element method, nonlinear equations

16 citations

Proceedings ArticleDOI
20 Jun 2022
TL;DR: In this paper , the authors proposed a method for computing turbulent solutions to Euler's equations with a slip boundary condition, which offers a Theory of Everything ToE for slightly viscous incompressible fluid flow as a parameter free model, covering a vast area of applications in vehicle aero/hydrodynamics including airplanes, ships and cars.
Abstract: We show that computing turbulent solutions to Euler’s equations with a slip boundary condition offers a Theory of Everything ToE for slightly viscous incompressible fluid flow as a parameter-free model, covering a vast area of applications in vehicle aero/hydrodynamics including airplanes, ships and cars. This work resolves the Grand Challenges of fluid dynamics described in NASA Vision 2030. The foundation of the methodology is an extremely efficient Direct FEM Simulation (DFS) method. We describe a breakthrough in efficiency, allowing extremely small numerical dissipation by choosing very small stabilization coefficients, while allowing very large time step size. This work is developed as part of the Digital Math framework - as the foundation of modern science based on constructive digital mathematical computation. We invite you to run and modify the simulations yourself in your web browser. The Digital Math web environment with the Open Source Real Flight Simulator/FEniCS software for reproducing the results in the paper at in principle "zero" cost, together with more detailed presentation and results is available at: http://digitalmath.tech/hiliftpw4-aiaa We show that Euler CFD by the scientific method in Digital Math predicts drag, lift and pressure distribution in close correspondence with observations for real problems with complex geometry with specific focus on the 4th High Lift Prediction Workshop and so can serve to deliver complete realistic aero/hydro-data for simulators without input from model experiments in wind tunnel and towing tank or full-scale experiments, as a new revolutionary capability.

3 citations

Peer Review
30 Dec 2022
TL;DR: In this paper , the Stokes paradox was studied from three different points of view: modern functional analysis, numerical simulations, and classical analytic techniques, and it was shown that the paradox still holds for the Reynolds-Orr equations describing kinetic energy instability, meaning that the instability steadily increases with domain size.
Abstract: Stokes flow equations, used to model creeping flow, are a commonly used simplification of the Navier–Stokes equations. The simplification is valid for flows where the inertial forces are negligible compared to the viscous forces. In infinite domains, this simplification leads to a fundamental paradox. In this work we review the Stokes paradox and present new insights related to recent research. We approach the paradox from three different points of view: modern functional analysis, numerical simulations, and classical analytic techniques. The first approach yields a novel, rigorous derivation of the paradox. We also show that relaxing the Stokes no-slip condition (by introducing a Navier’s friction condition) in one case resolves the Stokes paradox but gives rise to d’Alembert’s paradox. The Stokes paradox has previously been resolved by Oseen, who showed that it is caused by a limited validity of Stokes’ approximation. We show that the paradox still holds for the Reynolds–Orr equations describing kinetic energy flow instability, meaning that flow instability steadily increases with domain size. We refer to this as an instability paradox.

1 citations


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Book
01 Jan 2000
TL;DR: In this paper, a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics is presented, focusing on methods for linear elliptic boundary value problems.
Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

2,607 citations

Journal ArticleDOI
TL;DR: In this article, a modified Lagrange type interpolation operator is proposed to approximate functions in Sobolev spaces by continuous piecewise polynomials, and the combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.
Abstract: In this paper, we propose a modified Lagrange type interpolation operator to approximate functions in Sobolev spaces by continuous piecewise polynomials. In order to define interpolators for "rough" functions and to preserve piecewise polynomial boundary conditions, the approximated functions are averaged appropriately either on dor (d 1)-simplices to generate nodal values for the interpolation operator. This combination of averaging and interpolation is shown to be a projection, and optimal error estimates are proved for the projection error.

1,648 citations

Journal ArticleDOI
TL;DR: In this article, two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces.
Abstract: Two families of mixed finite elements, one based on triangles and the other on rectangles, are introduced as alternatives to the usual Raviart-Thomas-Nedelec spaces. Error estimates inL 2 (Ω) andH ?5 (Ω) are derived for these elements. A hybrid version of the mixed method is also considered, and some superconvergence phenomena are discussed.

1,213 citations

BookDOI
01 Jan 1990

1,149 citations