Optimal anisotropic meshes for minimizing interpolation errors in $L^p$-norm

Чэн, Лонг; Sun, Pengtao; Xu, Jinchao

doi:10.1090/s0025-5718-06-01896-5

Cited by 125 publications

(141 citation statements)

References 74 publications

(67 reference statements)

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“…This error bound extends the optimal interpolation error estimates for linear elements in [2,10,11,16] to higher order elements in R 2 . The above conclusions also agree with those based on the exact error formulas in the model problems of linear interpolation of a quadratic function (k = 1) and quadratic interpolation of a cubic function (k = 2) presented in [8] and [9], respectively.…”

Section: Introductionsupporting

confidence: 57%

“…The main conclusion is: given the area of a general triangular element τ , the error (in various norms) for the linear interpolation of a function u at the vertices of τ is nearly the minimum when τ is aligned with the eigenvector (associated with the smaller eigenvalue) of the Hessian ∇ 2 u, and the aspect ratio (or stretch ratio) of τ is about the square root of the ratio of the larger eigenvalue of ∇ 2 u to the smaller one. The globally optimal or nearly optimal mesh can be further characterized by the equidistribution of the interpolation error over each element [17,10,11].…”

Section: Introductionmentioning

confidence: 99%

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An interpolation error estimate in $\mathcal{R}^2$ based on the anisotropic measures of higher order derivatives

Cao

2008

Math. Comp.

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Abstract. In this paper, we introduce the magnitude, orientation, and anisotropic ratio for the higher order derivative ∇ k+1 u (with k ≥ 1) of a function u to characterize its anisotropic behavior. The magnitude is equivalent to its usual Euclidean norm. The orientation is the direction along which the absolute value of the k + 1-th directional derivative is about the smallest, while along its perpendicular direction it is about the largest. The anisotropic ratio measures the strength of the anisotropic behavior of ∇ k+1 u. These quantities are invariant under translation and rotation of the independent variables. They correspond to the area, orientation, and aspect ratio for triangular elements. Based on these measures, we derive an anisotropic error estimate for the piecewise polynomial interpolation over a family of triangulations that are quasi-uniform under a given Riemannian metric. Among the meshes of a fixed number of elements it is identified that the interpolation error is nearly the minimum on the one in which all the elements are aligned with the orientation of ∇ k+1 u, their aspect ratios are about the anisotropic ratio of ∇ k+1 u, and their areas make the error evenly distributed over every element.

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Section: Introductionsupporting

confidence: 57%

Section: Introductionmentioning

confidence: 99%

An interpolation error estimate in $\mathcal{R}^2$ based on the anisotropic measures of higher order derivatives

Cao

2008

Math. Comp.

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“…Graded meshes can be generated by considering the interpolation error to an arbitrary convex function f in (2.1) with the Hessian of f serving as a metric [4]. However, this is rather restrictive, because in general there does not exist a function whose Hessian matches a given arbitrary Riemannian metric.…”

Section: Weighted Odtmentioning

confidence: 99%

“…An example of the sizing function is shown in Figure 13, where blue and red indicate the minimum and maximum values of the sizing function. Then a density function is defined as 1/μ 5 , according to the fact that ODTs tend to equidistribute the weighted volume τ ρ d d+2 dx [4]. The density function used in the restricted CVT method for boundary surface remeshing is set to 1/μ 4 .…”

Section: Sliver-free Graded Meshmentioning

confidence: 99%

Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation

Chen¹,

Wang²,

Lévy³

et al. 2014

SIAM J. Sci. Comput.

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Abstract. This paper proposes a new algorithm to generate a graded three-dimensional tetrahedral mesh. It revisits the class of methods based on optimal Delaunay triangulation (ODT) and proposes a proper way of injecting a background density function into the objective function minimized by ODT. This continuous/analytic point of view leads to an objective function that is continuous and Delaunay consistent, in contrast with the discrete/geometrical point of view developed in previous work. To optimize the objective function, this paper proposes a hybrid algorithm that combines a local search (quasi-Newton) with a global optimization (simulated annealing). The benefits of the method are both improved performances and an improved quality of the result in terms of dihedral angles. This results from the combination of two effects. First, the local search has a faster speed of convergence than previous work due to the better behavior of the objective function, and second, the algorithm avoids getting stuck in a poor local minimum. Experimental results are evaluated and compared using standard metrics.

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“…While an almost best approximation property of finiteelement solutions in the maximum norm has been rigourously proved (with a logarithmic factor in the case of linear elements) for some equations on quasi-uniform meshes [12,13], there is no such result for strongly-anisotropic triangulations. Nevertheless, this perception is frequently considered a reasonable heuristic conjecture to be used in the anisotropic mesh adaptation [7,6,8,4].…”

Section: Introductionmentioning

confidence: 99%

Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations

Kopteva

2014

Math. Comp.

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Abstract. We give a counterexample of an anisotropic triangulation on which the exact solution has a second-order error of linear interpolation, while the computed solution obtained using linear finite elements is only first-order pointwise accurate. Our example is given in the context of a singularly perturbed reaction-diffusion equation, whose exact solution exhibits a sharp boundary layer. Furthermore, we give a theoretical justification of the observed numerical phenomena using a finite-difference representation of the considered finite element methods. Both standard and lumped-mass cases are addressed.

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Optimal anisotropic meshes for minimizing interpolation errors in $L^p$-norm

Cited by 125 publications

References 74 publications

An interpolation error estimate in $\mathcal{R}^2$ based on the anisotropic measures of higher order derivatives

An interpolation error estimate in $\mathcal{R}^2$ based on the anisotropic measures of higher order derivatives

Revisiting Optimal Delaunay Triangulation for 3D Graded Mesh Generation

Linear finite elements may be only first-order pointwise accurate on anisotropic triangulations

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