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73
Quasioptimal convergence rate for an adaptive finite element method
, 2007
"... We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refineme ..."
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Cited by 102 (15 self)
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We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
An adaptive finite element method for the Laplace–Beltrami operator on implicitly defined surfaces
 SIAM J. Numer. Anal
"... Abstract. We present an adaptive finite element method for approximating solutions to the LaplaceBeltrami equation on surfaces in R3 which may be implicitly represented as level sets of smooth functions. Residualtype a posteriori error bounds which show that the error may be split into a “residual ..."
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Cited by 49 (4 self)
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Abstract. We present an adaptive finite element method for approximating solutions to the LaplaceBeltrami equation on surfaces in R3 which may be implicitly represented as level sets of smooth functions. Residualtype a posteriori error bounds which show that the error may be split into a “residual part ” and a “geometric part ” are established. In addition, implementation issues are discussed and several computational examples are given. Key words. LaplaceBeltrami operator, adaptive finite element methods, a posteriori error estimation, boundary value problems on surfaces AMS subject classification. 58J32, 65N15, 65N30 1. Introduction. In
The finite element approximation of the nonlinear poissonboltzmann equation
 SIAM Journal on Numerical Analysis
"... ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible t ..."
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Cited by 33 (13 self)
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ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson–Boltzmann equation based on certain quasiuniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson–Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson–Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson– Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.
Quasioptimal convergence rate of an adaptive discontinuous Galerkin method
 SIAM J. NUMER. ANAL
, 2010
"... We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction ..."
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Cited by 20 (4 self)
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We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension ≥ 2. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded, and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental to derive optimal cardinality of ADFEM. We show that ADFEM (and AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
Convergence and optimality of adaptive mixed finite element methods
, 2009
"... The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved ..."
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Cited by 20 (6 self)
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The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasiorthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation.
Parametric FEM for Geometric Biomembranes
, 2011
"... We consider geometric biomembranes governed by an L²gradient flow for bending energy subject to area and volume constraints (Helfrich model). We give a concise derivation of a novel vector formulation, based on shape differential calculus, and corresponding discretization via parametric FEM using q ..."
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Cited by 17 (4 self)
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We consider geometric biomembranes governed by an L²gradient flow for bending energy subject to area and volume constraints (Helfrich model). We give a concise derivation of a novel vector formulation, based on shape differential calculus, and corresponding discretization via parametric FEM using quadratic isoparametric elements and a semiimplicit Euler method. We document the performance of the new parametric FEM with a number of simulations leading to dumbbell, red blood cell and toroidal equilibrium shapes while exhibiting large deformations.
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSONBOLTZMANN EQUATION
"... ABSTRACT. We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear PoissonBoltzmann equation (PBE). We first examine the twoterm regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of t ..."
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Cited by 15 (9 self)
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ABSTRACT. We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear PoissonBoltzmann equation (PBE). We first examine the twoterm regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the PoissonBoltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this twoterm regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L ∞ estimates to establish quasiorthogonality. To provide a highquality geometric model as input to the AFEM algorithm, we also describe a class of featurepreserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEM algorithm is also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.
Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains
 Electron. Trans. Numer. Anal
"... Abstract. We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain Ω that may have cracks or vertices that touch the boundary. We consider in ..."
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Cited by 15 (12 self)
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Abstract. We study theoretical and practical issues arising in the implementation of the Finite Element Method for a strongly elliptic second order equation with jump discontinuities in its coefficients on a polygonal domain Ω that may have cracks or vertices that touch the boundary. We consider in particular the equation − div(A∇u) = f ∈ H m−1 (Ω) with mixed boundary conditions, where the matrix A has variable, piecewise smooth coefficients. We establish regularity and Fredholm results and, under some additional conditions, we also establish wellposedness in weighted Sobolev spaces. When Neumann boundary conditions are imposed on adjacent sides of the polygonal domain, we obtain the decomposition u = ureg + σ, into a function ureg with better decay at the vertices and a function σ that is locally constant near the vertices, thus proving wellposedness in an augmented space. The theoretical analysis yields interpolation estimates that are then used to construct improved graded meshes recovering the (quasi)optimal rate of convergence for piecewise polynomials of degree m ≥ 1. Several numerical tests are included. Key words. NeumannNeumann vertex, transmission problem, augmented weighted Sobolev space, finite element method, graded mesh, optimal rate of convergence AMS subject classifications. 65N30, 35J25, 46E35, 65N12
Adaptive multiresolution analysis based on anisotropic triangulations, preprint, Laboratoire J.L.Lions, submitted 2008
"... A simple greedy refinement procedure for the generation of dataadapted triangulations is proposed and studied. Given a function f of two variables, the algorithm produces a hierarchy of triangulations (Dj)j≥0 and piecewise polynomial approximations of f on these triangulations. The refinement proce ..."
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Cited by 13 (5 self)
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A simple greedy refinement procedure for the generation of dataadapted triangulations is proposed and studied. Given a function f of two variables, the algorithm produces a hierarchy of triangulations (Dj)j≥0 and piecewise polynomial approximations of f on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between f and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the L p norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of f in case of C 2 functions). 1
ROBUST APOSTERIORI ESTIMATOR FOR ADVECTIONDIFFUSIONREACTION PROBLEMS
"... Abstract. We propose an almostrobust residualbased aposteriori estimator for the advectiondiffusionreaction model problem. The theory is developed in the onedimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays ..."
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Cited by 12 (0 self)
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Abstract. We propose an almostrobust residualbased aposteriori estimator for the advectiondiffusionreaction model problem. The theory is developed in the onedimensional setting. The numerical error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the energy norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful aposteriori estimator, robust up to a √ log(Pe) factor, where Pe is the global Péclet number of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature. We also consider a possible twodimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the onedimensional setting. 1.