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25
Metric tensors for anisotropic mesh generation
, 2005
"... It has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features. In practice, an anisotropic mesh is commonly gene ..."
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Cited by 39 (9 self)
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It has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features. In practice, an anisotropic mesh is commonly generated as a quasiuniform mesh in the metric determined by a tensor specifying the shape, size, orientation of elements. Thus, it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation. In this paper, we develop a general formula for the metric tensor for use in any spatial dimension. The formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements. Numerical results in twodimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality. The procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates.
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.
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.
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
A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates
 Journal of Computational Physics
"... Abstract. A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasiuniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error esti ..."
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Cited by 9 (3 self)
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Abstract. A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasiuniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric GaußSeidel method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients. 1.
Anisotropic measure of third order derivatives and the quadratic interpolation error on triangular elements
 SIAM J. Sci. Comput
"... Abstract. The main purpose of this paper is to present a closer look at how the H1 and L2errors for quadratic interpolation on a triangle are determined by the triangle geometry and the anisotropic behavior of the third order derivatives of interpolated functions. We characterize quantitatively th ..."
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Cited by 8 (2 self)
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Abstract. The main purpose of this paper is to present a closer look at how the H1 and L2errors for quadratic interpolation on a triangle are determined by the triangle geometry and the anisotropic behavior of the third order derivatives of interpolated functions. We characterize quantitatively the anisotropic behavior of a third order derivative tensor by its orientation and anisotropic ratio. Both exact error formulas and numerical experiments are presented for model problems of interpolating a cubic function u at the vertices and the midpoints of three sides of a triangle. Based on the study on model problems, we conclude that when an element is aligned with the orientation of ∇3u, the aspect ratio leading to nearly the smallest H1 and L2norms of the interpolation error is approximately equal to the anisotropic ratio of ∇3u. With this alignment and aspect ratio taken, the H1seminorm of the error is proportional to the reciprocal of the anisotropic ratio of ∇3u, the L2norm of the error is proportional to the − 3 th power of the anisotropic ratio, 2 and both of them are insensitive to the internal angles of the element. Key words. alignment anisotropic mesh, quadratic interpolation, anisotropic ratio, aspect ratio, mesh
An interpolation error estimate on anisotropic meshes in R n and optimal metrics for mesh refinement
 SIAM J. Numer. Anal
"... Abstract. In this paper, we extend the work in [W. Cao, Math. Comp., to appear] to functions of n dimensions. We measure the anisotropic behavior of higherorder derivative tensors by the “largest ” (in certain sense) ellipse/ellipsoid contained in the level curve/surface of the polynomial for direc ..."
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Cited by 7 (0 self)
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Abstract. In this paper, we extend the work in [W. Cao, Math. Comp., to appear] to functions of n dimensions. We measure the anisotropic behavior of higherorder derivative tensors by the “largest ” (in certain sense) ellipse/ellipsoid contained in the level curve/surface of the polynomial for directional derivatives. Given the anisotropic measure for the interpolated functions, we derive an error estimate for piecewise polynomial interpolations on meshes that are quasiuniform under a given metric. By using the inertia properties for matrix eigenvalues [R. C. Thompson, J. Math. Anal. Appl., 58 (1977), pp. 572–577] and Hölder’s inequality, we can identify the optimal mesh metrics leading to the smallest error bound in various norms. Furthermore, we develop a dimensional reduction method to find the anisotropic measure approximately. We present two numerical examples for linear and quadratic interpolation on various anisotropic meshes generated with the optimal mesh metrics developed in this paper. Numerical results show that the smallest interpolation error is attained exactly on meshes optimal for the corresponding error norm as predicted.
2 CONDITIONING OF FINITE ELEMENT EQUATIONS WITH ARBITRARY ANISOTROPIC MESHES
"... Abstract. Bounds are developed for the condition number of the linear system resulting from the finite element discretization of an anisotropic diffusion problem with arbitrary meshes. These bounds are shown to depend on three major factors: a factor representing the base order corresponding to the ..."
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Cited by 6 (4 self)
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Abstract. Bounds are developed for the condition number of the linear system resulting from the finite element discretization of an anisotropic diffusion problem with arbitrary meshes. These bounds are shown to depend on three major factors: a factor representing the base order corresponding to the condition number for a uniform mesh, a factor representing the effects of the mesh Mnonuniformity (mesh nonuniformity in the metric tensor defined by the diffusion matrix), and a factor representing the effects of the mesh volumenonuniformity. Diagonal scaling for the finite element linear system and its effects on the conditioning are studied. It is shown that a properly chosen diagonal scaling can eliminate the effects of the mesh volumenonuniformity and reduce the effects of the mesh Mnouniformity on the conditioning of the stiffness matrix. In particular, the bound after a proper diagonal scaling depends only on a volumeweighted average (instead of the maximum for the unscaled case) of a quantity measuring the mesh Mnonuniformity. Bounds on the extreme eigenvalues of the stiffness and mass matrices are also investigated. Numerical examples are presented to verify the theoretical findings. 4 5
Discontinuous Galerkin methods on hpanisotropic meshes II: A posteriori error analysis and adaptivity
 In preparation
"... We consider the a priori error analysis of hpversion interior penalty discontinuous Galerkin methods for second{order partial dierential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local nite element spaces employed. In particular, we prove a pr ..."
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Cited by 6 (0 self)
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We consider the a priori error analysis of hpversion interior penalty discontinuous Galerkin methods for second{order partial dierential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local nite element spaces employed. In particular, we prove a priori hperror bounds for linear target functionals of the solution, on (possibly) anisotropic computational meshes with anisotropic tensorproduct polynomial basis functions. The theoretical results are illustrated by a numerical experiment. 1
Adaptive Moving Mesh Modeling for Two Dimensional Groundwater Flow and Transport
 CONTEMPORARY MATHEMATICS
, 2004
"... An adaptive moving mesh method is presented for numerical simulation of two dimensional groundwater flow and transport problems. A selection of problems are considered, including advection dominated chemical transport and reaction, solute transport from contamination sources, transport of nonaqueou ..."
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Cited by 4 (1 self)
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An adaptive moving mesh method is presented for numerical simulation of two dimensional groundwater flow and transport problems. A selection of problems are considered, including advection dominated chemical transport and reaction, solute transport from contamination sources, transport of nonaqueous phase liquids (NAPLs) in an aquifer, and coupling of groundwater flow with NAPL transport. Numerical results show that the adaptive moving mesh method is able to capture sharp moving fronts and detect the emerging of new fronts.