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22
A Study Of Monitor Functions For TwoDimensional Adaptive Mesh Generation
 SIAM J. SCI. COMPUT
, 1999
"... In this paper we study the problem of twodimensional adaptive mesh generation using a variational approach and, specifically, the effect that the monitor function has on the resulting mesh behavior. The basic theoretical tools employed are Green's function for elliptic problems and the eigende ..."
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Cited by 55 (11 self)
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In this paper we study the problem of twodimensional adaptive mesh generation using a variational approach and, specifically, the effect that the monitor function has on the resulting mesh behavior. The basic theoretical tools employed are Green's function for elliptic problems and the eigendecomposition of symmetric positive definite matrices. Based upon this study, a general strategy is suggested for how to choose the monitor function, and numerical results are presented for illustrative purposes. The threedimensional case is also briefly discussed. It is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well.
Variational Mesh Adaptation: Isotropy and Equidistribution
 J. Comput. Phys
, 2001
"... We present a new approach for developing more robust and error oriented mesh adaptation methods. Speci cally, assuming that a regular (in cell shape) and uniform (in cell size) computational mesh is used (as is commonly done in computation), we develop a criterion for mesh adaptation based on an ..."
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Cited by 43 (11 self)
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We present a new approach for developing more robust and error oriented mesh adaptation methods. Speci cally, assuming that a regular (in cell shape) and uniform (in cell size) computational mesh is used (as is commonly done in computation), we develop a criterion for mesh adaptation based on an error function which de nition is motivated by the analysis of function variation and local error behavior for linear interpolation. The criterion is then decomposed into two aspects, the isotropy (or conformity) and uniformity (or equidistribution) requirements, each of which can be easier to deal with. The functionals that satisfy these conditions approximately are constructed using discrete and continuous inequalities. A new functional is nally formulated by combining the functionals corresponding to the isotropy and uniformity requirements. The features of the functional are analyzed and demonstrated by numerical results. In particular, unlike the existing mesh adaptation functionals, the new functional has clear geometric meanings of minimization. A mesh that has the desired properties of isotropy and equidistribution can be obtained by properly choosing the values of two parameters. The analysis presented in this article also provides a better understanding of the increasingly popular method of harmonic mapping in two dimensions.
Adaptivity with moving grids
, 2009
"... In this article we look at the modern theory of moving meshes as part of an radaptive strategy for solving partial differential equations with evolving internal structure. We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partia ..."
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Cited by 28 (5 self)
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In this article we look at the modern theory of moving meshes as part of an radaptive strategy for solving partial differential equations with evolving internal structure. We firstly examine the possible geometries of a moving mesh in both one and higher dimensions, and the discretization of partial differential equation on such meshes. In particular, we consider such issues as mesh regularity, equidistribution, variational methods, and the error in interpolating a function or truncation error on such a mesh. We show that, guided by these, we can design effective moving mesh strategies. We then look in more detail as to how these strategies are implemented. Firstly we look at positionbased methods and the use of moving mesh partial differential equation (MMPDE), variational and optimal transport methods. This is followed by an analysis of velocitybased methods such as the geometric conservation law (GCL) methods. Finally we look at a number of examples where the use of a moving mesh method is effective in applications. These include scaleinvariant problems, blowup problems, problems with moving fronts and problems in meteorology. We conclude that, whilst radaptive methods are still in a relatively new stage of development, with many outstanding questions remaining, they have enormous potential for development, and for many problems they represent an optimal form of adaptivity.
A moving mesh method based on the geometric conservation law
 SIAM Journal on Scientific Computing
"... Abstract. A new adaptive mesh movement strategy is presented, which, unlike many existing moving mesh methods, targets the mesh velocities rather than the mesh coordinates. The mesh velocities are determined in a least squares framework by using the geometric conservation law, specifying a form for ..."
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Cited by 25 (3 self)
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Abstract. A new adaptive mesh movement strategy is presented, which, unlike many existing moving mesh methods, targets the mesh velocities rather than the mesh coordinates. The mesh velocities are determined in a least squares framework by using the geometric conservation law, specifying a form for the Jacobian determinant of the coordinate transformation defining the mesh, and employing a curl condition. By relating the Jacobian to a monitor function, one is able to directly control the mesh concentration. The geometric conservation law, an identity satisfied by any nonsingular coordinate transformation, is an important tool which has been used for many years in the engineering community to develop cellvolumepreserving finitevolume schemes. It is used here to transform the algebraic expression specifying the Jacobian into an equivalent differential relation which is the key formula for the new mesh movement strategy. It is shown that the resulting method bears a close relation with the Lagrangian method. Advantages of the new approach include the ease of controlling the cell volumes (and therefore mesh adaption) and a theoretical guarantee for existence and nonsingularity of the coordinate transformation. It is shown that the method may suffer from the mesh skewness, a consequence resulting from its close relation with the Lagrangian method. Numerical results are presented to demonstrate various features of the new method.
Approaches for Generating Moving Adaptive Meshes: Location versus Velocity
 Appl. Numer. Math
, 2002
"... A variety of approaches for generating moving adaptive methods are summarized and compared. They basically fall in two groups: the velocity based methods and the location based ones. The features, including the advantage and weakness, of each group are addressed. ..."
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Cited by 24 (1 self)
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A variety of approaches for generating moving adaptive methods are summarized and compared. They basically fall in two groups: the velocity based methods and the location based ones. The features, including the advantage and weakness, of each group are addressed.
Mathematical principles of anisotropic mesh adaptation
 Commun. Comput. Phys
, 2006
"... Abstract. Mesh adaptation is studied from the mesh control point of view. Two principles, equidistribution and alignment, are obtained and found to be necessary and sufficient for a complete control of the size, shape, and orientation of mesh elements. A key component in these principles is the moni ..."
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Cited by 22 (6 self)
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Abstract. Mesh adaptation is studied from the mesh control point of view. Two principles, equidistribution and alignment, are obtained and found to be necessary and sufficient for a complete control of the size, shape, and orientation of mesh elements. A key component in these principles is the monitor function, a symmetric and positive definite matrix used for specifying the mesh information. A monitor function is defined based on interpolation error in a way with which an error bound is minimized on a mesh satisfying the equidistribution and alignment conditions. Algorithms for generating meshes satisfying the conditions are developed and twodimensional numerical results are presented.
Geometric Integration and Its Applications
 in Handbook of numerical analysis
, 2000
"... This paper aims to give an introduction to the relatively new eld of geometric integration. ..."
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Cited by 21 (2 self)
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This paper aims to give an introduction to the relatively new eld of geometric integration.
The Geometric Integration of Scale Invariant Ordinary and Partial Differential Equations
, 2000
"... This review paper examines a synthesis of adaptive mesh methods with the use of symmetry to solve ordinary and partial dierential equations. It looks at the eectiveness of numerical methods in preserving geometric structures of the underlying equations such as scaling invariance, conservation law ..."
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Cited by 16 (2 self)
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This review paper examines a synthesis of adaptive mesh methods with the use of symmetry to solve ordinary and partial dierential equations. It looks at the eectiveness of numerical methods in preserving geometric structures of the underlying equations such as scaling invariance, conservation laws and solution orderings. Studies are made of a series of examples including the porous medium equation and the nonlinear Schrodinger equation. key words: Mesh adaption, selfsimilar solution, scaling invariance, conservation laws, maximum principles, equidistribution. March 31, 2000 1 Dept. of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. cjb@maths.bath.ac.uk 2 Dept. of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. mapmdp@maths.bath.ac.uk 1 1 Introduction When we wish to nd a numerical approximation to the solution of a partial dierential equation, a natural technique is to discretise the PDE so that local trunc...
A Twodimensional Moving Finite Element Method With Local Refinement Based On A Posteriori Error Estimates
 Applied Numer. Math
"... In this paper, we consider the numerical solution of time{dependent PDEs using a nite element method based upon rh{adaptivity. An adaptive horizontal method of lines strategy equipped with a posteriori error estimates to control the discretization through variable time steps and spatial grid adaptat ..."
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Cited by 12 (2 self)
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In this paper, we consider the numerical solution of time{dependent PDEs using a nite element method based upon rh{adaptivity. An adaptive horizontal method of lines strategy equipped with a posteriori error estimates to control the discretization through variable time steps and spatial grid adaptations is used. Our approach combines an r{re nement method based upon solving so{called moving mesh PDEs with h{re nement. Numerical results are presented to demonstrate the capabilities and bene ts of combining mesh movement and local re nement.
Moving mesh generation using the Parabolic MongeAmpère equation
, 2008
"... This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear pa ..."
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Cited by 10 (4 self)
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This article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations in several spatial dimensions. The mesh is obtained by taking the gradient of a (scalar) mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation. This method gives a new technique for performing radaptivity based on ideas from Optimal Transportation combined with the equidistribution principle applied to a (time varying) scalar monitor function (used successfully in moving mesh methods in onedimension). Detailed analysis of this new method is presented in which the convergence, regularity and stability of the mesh is studied. Additionally, this new method is shown to be straightforward to program and implement, requiring the solution of only one simple scalar timedependent equation in arbitrary dimension, with adaptivity along the boundaries handled automatically. We discuss three preexisting methods in the context of this work. Examples are presented in which either the monitor function is prescribed in advance, or it is given by the solution of a partial