AFEM for Geometric PDE: The Laplace-Beltrami Operator

Bonito, Andrea; Cascón, J.M.; Morín, Pedro; Nochetto, Ricardo H.

doi:10.1007/978-88-470-2592-9_15

Cited by 12 publications

(32 citation statements)

References 60 publications

(114 reference statements)

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“…Values of ) with different N and the corresponding convergence rates are presented in Table 5. We can obtain from Table 5 that the convergence rate is approximately 0.4, which is as good a result as shown for the same example in [36], in which an adaptive finite element method (AFEM) is presented for several applications governed by geometric PDEs. Similar methods can be easily applied to numerically approximate the solutions of more complex problems.…”

Section: Graph: Conforming Patches and Same Degree Across The Patchessupporting

confidence: 73%

Discontinuous Galerkin Methods for Isogeometric Analysis for Elliptic Equations on Surfaces

Zhang

Chen

2014

Commun. Math. Stat.

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We propose a method that combines isogeometric analysis (IGA) with the discontinuous Galerkin (DG) method for solving elliptic equations on 3-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted across the patch interfaces to glue the multiple patches, while the traditional IGA, which is very suitable for solving partial differential equations (PDEs) on (3D) surfaces, is employed within each patch. Our method takes advantage of both IGA and the DG method. Firstly, the time-consuming steps in mesh generation process in traditional finite element analysis (FEA) are no longer necessary and refinements, including h-refinement and p-refinement which both maintain the original geometry, can be easily performed by knot insertion and order-elevation (Farin, in Curves and surfaces for CAGD, 2002). Secondly, our method can easily handle the cases with non-conforming patches and different degrees across the patches. Moreover, due to the geometric flexibility of IGA basis functions, especially the use of multiple patches, we can get more accurate modeling of more complex surfaces. Thus, the geometrical error is significantly reduced and it is, in particular, eliminated for all conic sections. Finally, this method can be easily formulated and implemented. We generally describe the problem and then present our primal formulation. A new ideology, which directly makes use of the approximation property of the NURBS basis functions on the parametric domain rather than that of the IGA functions on the physical domain (the former is easier to get), is adopted when we perform the theoretical analysis including the boundedness and stability of the primal form, and the error analysis under both the DG norm and the L 2 norm. The result of the error analysis shows that our scheme achieves the optimal convergence rate with respect to both the DG norm and the L 2 norm. Numerical examples are presented to verify the theoretical result and gauge the good performance of our method.

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Section: Graph: Conforming Patches and Same Degree Across The Patchessupporting

confidence: 73%

Discontinuous Galerkin Methods for Isogeometric Analysis for Elliptic Equations on Surfaces

Zhang

Chen

2014

Commun. Math. Stat.

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“…and they are less sensitive to mesh tangling due to tangential node motion for time dependent problems; we refer to [6] for a discussion of several applications. The advantage of high-order methods is even more pronounced when they are combined with adaptivity.…”

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confidence: 99%

“…AFEMs are known to exploit the nonlinear Besov regularity scale, instead of the linear Sobolev scale, and to deliver optimal convergence rates N −n/d in terms of degrees of freedom N for singular elliptic problems on flat domains with limited Sobolev regularity [30,12], [11,13,21,22,28]. The study of AFEM for the Laplace-Beltrami operator on parametric surfaces is, however, restricted to n = 1 because the first fundamental form, area element, and normal vector to the discrete surface as well as the surface gradient of discrete functions are piecewise constant, which greatly simplifies the analysis [6]. This paper bridges this gap and provides a comprehensive approach to high-order AFEM on parametric surfaces.…”

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confidence: 99%

“…finds adaptively a refinement T * of T such that the Galerkin solution U * ∈ V(T * ) on Γ * -the approximate surface corresponding to T * -satisfies the prescribed bound η T * (U * , F * ) ≤ ε. This is the usual loop for linear elliptic PDE [19,26], [6,8,11,12,13,22,25,28] SOLVE → ESTIMATE → MARK → REFINE, (1.6) except that the approximate surface Γ is updated after each REFINE call and therefore changes within ADAPT PDE.…”

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confidence: 99%

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High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates

et al. 2016

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We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally W 1 ∞ and piecewise in a suitable Besov class embedded in C 1,α with α ∈ (0, 1]. The idea is to have the surface sufficiently well resolved in W 1 ∞ relative to the current resolution of the PDE in H 1 . This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in W 1 ∞ and PDE error in H 1 .1. Introduction. Let γ be a d dimensional surface in R d+1 (d ≥ 1) either with or without boundary, which is globally Lipschitz and piecewise in a suitable Besov class embedded in C 1,α with α ∈ (0, 1]. We design and study a quasi-optimal adaptive finite element method (AFEM) to approximate the solution of( 1.1) where f ∈ L 2 (γ) and −∆ γ is the Laplace-Beltrami operator (or surface Laplacian) on γ. In addition, we impose that u = 0 on ∂γ or require that γ u = 0 if ∂γ = ∅ (with γ f = 0 for compatibility). To represent ∆ γ , one needs to describe γ mathematically using, for example, parametric representations on charts, level sets, distance functions, graphs of functions, etc. Moreover, one usually obtains approximate solutions (finite element solutions) by solving the problem on approximate polyhedral surfaces rather than the surface γ itself. Exploiting the variational structure of the Laplace-Beltrami operator, [20] gives an a priori error analysis whereas [17,16,25,6] provide a posteriori counterparts. Our present objective is to continue our research on AFEM for elliptic PDEs on surfaces initiated in [25] for graphs and extended in [6] to parametric surfaces, the latter with polynomial degree n = 1. We design herein an AFEM for parametric surfaces using C 0 finite elements of degree n ≥ 1, prove optimal convergence rates and workload estimates, and study suitable approximation classes for the triple (u, f, γ).High-order finite elements are superior to linears for geometric problems: they provide better approximation of important geometric quantities such as curvature,

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“…To quantify the deviation of a curved boundary face from its polygonal interpolation, we define the geometric element indicator, cf. [1],…”

Section: A Posteriori Error Analysismentioning

confidence: 99%

Adaptive FEM and a posteriori error control for stationary coupled bulk‐surface PDEs

Eigel

Müller

2016

Proc Appl Math and Mech

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We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. The numerical error of the finite element solution can be controlled by a residual a posteriori error estimator which takes into account the approximation errors due to the discretisation in space as well as the polyhedral approximation of the surface. The estimators naturally lead to refinement indicators for an adaptive algorithm to control the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm.

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AFEM for Geometric PDE: The Laplace-Beltrami Operator

Cited by 12 publications

References 60 publications

Discontinuous Galerkin Methods for Isogeometric Analysis for Elliptic Equations on Surfaces

Discontinuous Galerkin Methods for Isogeometric Analysis for Elliptic Equations on Surfaces

High-Order AFEM for the Laplace–Beltrami Operator: Convergence Rates

Adaptive FEM and a posteriori error control for stationary coupled bulk‐surface PDEs

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