Moving Finite Elements. I

Miller, Keith; Miller, Robert N.

doi:10.1137/0718070

Cited by 439 publications

(192 citation statements)

References 4 publications

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“…One method of adapting the mesh to the solution of an evolution equation is to let the mesh evolve with the solution [5,19,20]. For convection dominated problems it is natural to let the mesh points flow along the streamlines (characteristics) of (an approximation of) the velocity.…”

Section: Moving Meshesmentioning

confidence: 99%

Lagrangian and moving mesh methods for the convection diffusion equation

Chrysafinos¹,

Walkington²

2008

ESAIM: M2AN

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Section: Moving Meshesmentioning

confidence: 99%

Lagrangian and moving mesh methods for the convection diffusion equation

Chrysafinos¹,

Walkington²

2008

ESAIM: M2AN

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“…There are several methods to obtain a moving mesh (see e.g. [1,3,22,33,37,39,40]). Here, we adopt the so-called moving mesh PDE approach [31][32][33] in which a time-dependent PDE is introduced to determine the motion of the mesh.…”

Section: The Moving Meshmentioning

confidence: 99%

“…Mesh adaptivity is usually in the form of local mesh refinements or through a bijective and continuous mesh mapping. The adaptive mesh can also be static or dynamic (continuously moving) [1,3,22,33,37,39,40,44]. In local adaptive mesh refinement methods (see e.g.…”

Section: Introductionmentioning

confidence: 99%

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Ceniceros¹,

Hou²

2001

Journal of Computational Physics

147

156

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We develop an efficient dynamically adaptive mesh generator for time-dependent problems in two or more dimensions. The mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic PDEs for the mesh map. When coupled to a physical problem, the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior. The overall mesh strategy is simple to implement, avoids interpolation, and can be easily incorporated into a broad range of applications. The efficacy of the mesh is first demonstrated by two examples of blowing-up solutions to the 2-D semilinear heat equation. These examples show that the mesh can follow with high adaptivity a finite-time singularity process. The focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2-D Boussinesq fluid, a challenging problem. The moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically. The numerical results show the fast collapse to small scales and an exponential vorticity growth.

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“…moving finite element methods [2], geometric conservation law methods [3]), and location based methods which directly control the mesh points themselves (e.g. equidistribution based methods [4,5], harmonic mapping [6]).…”

Section: Introductionmentioning

confidence: 99%

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

Ramage¹,

Newton²

2008

Molecular Crystals and Liquid Crystals

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The Strathprints institutional repository (https://strathprints.strath.ac.uk) is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, expose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output. AbstractThis paper illustrates the use of moving mesh methods for solving partial differential equation (PDE) problems in Q-tensor theory of liquid crystals. We present the results of an initial study using a simple one-dimensional test problem which illustrates the feasibility of applying adaptive grid techniques in such situations. We describe how the grids are computed using an equidistribution principle, and investigate the comparative accuracy of adaptive and uniform grid strategies, both theoretically and via numerical examples.

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Moving Finite Elements. I

Cited by 439 publications

References 4 publications

Lagrangian and moving mesh methods for the convection diffusion equation

Lagrangian and moving mesh methods for the convection diffusion equation

An Efficient Dynamically Adaptive Mesh for Potentially Singular Solutions

Adaptive Grid Methods forQ-Tensor Theory of Liquid Crystals: A One-Dimensional Feasibility Study

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