Design and Application of a Gradient-Weighted Moving Finite Element Code I: in One Dimension

Carlson, Neil N.; Miller, Keith

doi:10.1137/s106482759426955x

Cited by 125 publications

(99 citation statements)

References 21 publications

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“…Let u h be a moving grid finite element approximation to u defined by 6) where ξ(t) represents the finite element moving grid. Then the moving grid finite element approximation seeks to find u h ∈ V h ⊂ H 1 such that…”

Section: The Moving Grid Finite Element Methodsmentioning

confidence: 99%

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

2005

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Numerical techniques for moving meshes are many and varied. In this paper we present a novel application of a moving grid finite element method applied to biological problems related to pattern formation where the mesh movement is prescribed through a specific definition to mimic the growth that is observed in nature. Through the use of a moving grid finite element technique, we present numerical computational results illustrating how period doubling behaviour occurs as the domain doubles in size.

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Section: The Moving Grid Finite Element Methodsmentioning

confidence: 99%

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

2005

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“…Burgers' equation is similar in form to the linear advection equation, except that the convective term is non-linear thus allowing discontinuous solutions to develop. The initial and boundary conditions considered for this standard test were the same as in [2]: uðx; 0Þ ¼ sin 2px ð Þþsin px ð Þ=2; uð0; tÞ ¼ uð1; tÞ ¼ 0:…”

Section: Burgers' Equationmentioning

confidence: 99%

Adaptive multiresolution approach for solution of hyperbolic PDEs

Alves

Cruz

Mendes

et al. 2002

Computer Methods in Applied Mechanics and Engineering

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This paper establishes an innovative and efficient multiresolution adaptive approach combined with high-resolution methods, for the numerical solution of a single or a system of partial differential equations. The proposed methodology is unconditionally bounded (even for hyperbolic equations) and dynamically adapts the grid so that higher spatial resolution is automatically allocated to domain regions where strong gradients are observed, thus possessing the two desired properties of a numerical approach: stability and accuracy.Numerical results for five test problems are presented which clearly show the robustness and cost effectiveness of the proposed method.

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“…Moving-mesh methods consist in moving the nodes of a given initial mesh to new locations, the number and connectivity of nodes being fixed (see [5] and references therein). For steady-state problems in which the true solution minimizes a functional, the idea is to look for the nodes that minimize the functional obtained by a finite element discretization [10,14] ("moving finite elements").…”

Section: E(θ) := E(u)mentioning

confidence: 99%

“…This tangling difficulty is well-known in the context of moving-mesh methods. It can be cured by the introduction of internodal forces [5]. Here, such methods are not necessary.…”

Section: Computational Considerationsmentioning

confidence: 99%

Moving mesh for the axisymmetric harmonic map flow

Merlet¹,

Pierre

2005

ESAIM: M2AN

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Abstract. We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L 2 -gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method.Mathematics Subject Classification. 35A05, 35K55, 65N30, 65N50, 65N99.

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Design and Application of a Gradient-Weighted Moving Finite Element Code I: in One Dimension

Cited by 125 publications

References 21 publications

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

Adaptive multiresolution approach for solution of hyperbolic PDEs

Moving mesh for the axisymmetric harmonic map flow

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