The group finite element formulation

Fletcher, Christopher W.

doi:10.1016/0045-7825(83)90122-6

Cited by 106 publications

(57 citation statements)

References 21 publications

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“…Surprisingly enough, it was also found to produce a small gain in accuracy e.g. when applied to the Burgers equation on a uniform grid [24]. The resulting savings in CPU time become increasingly pronounced for multidimensional problems and=or strong non-linearities.…”

Section: Galerkin Flux Decompositionmentioning

confidence: 94%

“…Therefore, the positivity of u n is inherited by u n+1 provided that the time step is small enough. According to (24), the backward Euler time stepping is unconditionally positive. The Crank-Nicolson scheme is subject to the CFL-like condition for the auxiliary problem (50), but the admissible Courant numbers are twice as large as those for the fully explicit scheme.…”

Section: Positivity Proofmentioning

confidence: 99%

“…This approach termed the group ÿnite-element formulation by Fletcher [24] provides an e cient matrix assembly leading to a considerable reduction in the computational cost. Surprisingly enough, it was also found to produce a small gain in accuracy e.g.…”

Section: Galerkin Flux Decompositionmentioning

confidence: 99%

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Multidimensional FEM‐FCT schemes for arbitrary time stepping

Kuzmin

Möller

Turek

2003

Numerical Methods in Fluids

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SUMMARYThe ux-corrected-transport paradigm is generalized to ÿnite-element schemes based on arbitrary time stepping. A conservative ux decomposition procedure is proposed for both convective and di usive terms. Mathematical properties of positivity-preserving schemes are reviewed. A nonoscillatory loworder method is constructed by elimination of negative o -diagonal entries of the discrete transport operator. The linearization of source terms and extension to hyperbolic systems are discussed. Zalesak's multidimensional limiter is employed to switch between linear discretizations of high and low order. A rigorous proof of positivity is provided. The treatment of non-linearities and iterative solution of linear systems are addressed. The performance of the new algorithm is illustrated by numerical examples for the shock tube problem in one dimension and scalar transport equations in two dimensions.

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Section: Galerkin Flux Decompositionmentioning

confidence: 94%

Section: Positivity Proofmentioning

confidence: 99%

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Multidimensional FEM‐FCT schemes for arbitrary time stepping

Kuzmin

Möller

Turek

2003

Numerical Methods in Fluids

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“…This approach is motivated by results that show that for the model Burger's equation this grouped discretization yields slightly higher accuracy than the ungrouped scheme [21]. This approach is one of the several alternatives presented by Morgan and Peraire for the Galerkin finite element method with the explicit addition of diffusion [22].…”

Section: Finite Element Formulationmentioning

confidence: 95%

“…Matrices A i and K i j are both functions of the independent variables U and are listed explicitly in Reference [2]. Using (21) and (22) in (18) yields the second-order system…”

Section: System Of Equationsmentioning

confidence: 99%

Development and validation of a SUPG finite element scheme for the compressible Navier–Stokes equations using a modified inviscid flux discretization

Kirk

Carey

2007

Numerical Methods in Fluids

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SUMMARYThis paper considers the streamline-upwind Petrov-Galerkin (SUPG) method applied to the unsteady compressible Navier-Stokes equations in conservation-variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, and the second-order accurate time discretization are presented. The numerical method is discussed in detail. The performance of the algorithm is then investigated by considering inviscid flow past a circular cylinder. Validation of the finite element formulation via comparisons with experimental data for high-Mach number perfect gas laminar flows is presented, with a specific focus on comparisons with experimentally measured skin friction and convective heat transfer on a 15 • compression ramp.

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A density‐dependent FEM‐FCT algorithm with application to modeling platelet aggregation

Danes

Leiderman

2019

Numer Methods Biomed Eng

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Upon injury to a blood vessel, flowing platelets will aggregate at the injury site, forming a plug to prevent blood loss. Through a complex system of biochemical reactions, a stabilizing mesh forms around the platelet aggregate forming a blood clot that further seals the injury. Computational models of clot formation have been developed to a study intravascular thrombosis, where a vessel injury does not cause blood leakage outside the blood vessel but blocks blood flow. To model scenarios in which blood leaks from a main vessel out into the extravascular space, new computational tools need to be developed to handle the complex geometries that represent the injury. We have previously modeled intravascular clot formation under flow using a continuum approach wherein the transport of platelet densities into some spatial location is limited by the platelet fraction that already reside in that location, i.e., the densities satisfy a maximum packing constraint through the use of a hindered transport coefficient. To extend this notion to extravascular injury geometries, we have modified a finite element method flux‐corrected transport (FEM‐FCT) scheme by prelimiting antidiffusive nodal fluxes. We show that our modified scheme, under a variety of test problems, including mesh refinement, structured vs unstructured meshes, and for a range of reaction rates, produces numerical results that satisfy a maximum platelet‐density packing constraint.

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The group finite element formulation

Cited by 106 publications

References 21 publications

Multidimensional FEM‐FCT schemes for arbitrary time stepping

Multidimensional FEM‐FCT schemes for arbitrary time stepping

Development and validation of a SUPG finite element scheme for the compressible Navier–Stokes equations using a modified inviscid flux discretization

A density‐dependent FEM‐FCT algorithm with application to modeling platelet aggregation

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