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272
Grid adaptation for functional outputs: application to twodimensional inviscid flows
 J. Comput. Phys
"... www.elsevier.com/locate/jcp ..."
A Posteriori Finite Element Bounds for LinearFunctional Outputs of Elliptic Partial Differential Equations
 Computer Methods in Applied Mechanics and Engineering
, 1997
"... We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order elliptic linear partial differential equations in two space dimensions. The method is base ..."
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Cited by 46 (9 self)
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We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order elliptic linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic "energy" reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine "truthmesh" discretization are then derived by appealing to a dual maxmin relaxation evaluated for optimally chosen adjoint and hybridflux candidate Lagrange multipliers generated by a Kelement coarser "workingmesh" approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomainlocal, symmetric Neumann pro...
Convergence of adaptive finite element methods
 SIAM Review
"... Abstract. We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasiorthogonality, which accounts for the bilinear form not being a scalar product, together ..."
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Cited by 43 (5 self)
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Abstract. We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasiorthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEM is a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and convectiondiffusion PDE, illustrate the theory and yield optimal meshes.
Optimality of a standard adaptive finite element method
"... In this paper, an adaptive ¯nite element method is constructed
for solving elliptic equations that has optimal computational complexity.
Whenever for some s > 0, the solution can be approximated to accuracy
O(n¡s) in energy norm by a continuous piecewise linear function on some
partition with n tria ..."
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Cited by 40 (3 self)
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In this paper, an adaptive ¯nite element method is constructed
for solving elliptic equations that has optimal computational complexity.
Whenever for some s > 0, the solution can be approximated to accuracy
O(n¡s) in energy norm by a continuous piecewise linear function on some
partition with n triangles, and one knows how to approximate the righthand
side in the dual norm with the same rate with piecewise constants, then
the adaptive method produces approximations that converge with this rate,
taking a number of operations that is of the order of the number of triangles
in the output partition. The method is similar in spirit to that from [SINUM,
38 (2000), pp.466{488] by Morin, Nochetto, and Siebert, and so in particular
it does not rely on a recurrent coarsening of the partitions. Although the
Poisson equation in two dimensions with piecewise linear approximation is
considered, it can be expected that the results generalize in several respects.
GoalOriented Error Estimation and Adaptivity for the Finite Element Method
 Comput. Math. Appl
, 1999
"... this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These socalled quantities of interest are characterized by linear functionals on ..."
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Cited by 39 (1 self)
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this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These socalled quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundaryvalue problems, and in particular, show how to obtain accurate estimates as well as upper and lowerbounds on the error. We also study the new concept of goaloriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.
Adjoint Error Estimation and Grid Adaptation for Functional Outputs: Application to QuasiOneDimensional Flow
 J. Comput. Phys
, 2000
"... this paper, attention is limited to onedimensional problems, although the procedure is readily extendable to multiple dimensions. The error estimation procedure is applied to a standard, secondorder, finite volume discretization of the quasionedimensional Euler equations. Both isentropic and ..."
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Cited by 39 (8 self)
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this paper, attention is limited to onedimensional problems, although the procedure is readily extendable to multiple dimensions. The error estimation procedure is applied to a standard, secondorder, finite volume discretization of the quasionedimensional Euler equations. Both isentropic and shocked flows are considered. The chosen functional of interest is the integrated pressure along a variablearea duct. The error estimation procedure, applied on uniform grids, provides superconvergent values of the corrected functional. Results demonstrate that additional improvements in the accuracy of the functional can be achieved by applying the proposed adaptive strategy to an initially uniform grid
Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids
 Part II: Higher order FEM, Math. Comp., posted on February 4, 2002, PII S
"... Abstract. Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conformi ..."
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Cited by 35 (8 self)
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Abstract. Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given righthand sides. 1.
Bounds for LinearFunctional Outputs of Coercive Partial Differential Equations: Local Indicators and Adaptive Refinement
, 1997
"... this paper we focus on three new developments. First, we introduce a modified energy objective, and hence modified Lagrangian, that permits both more transparent interpretation and more ready generalization. Second, we demonstrate that the bound gap  the difference between the upper and lower bou ..."
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Cited by 33 (9 self)
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this paper we focus on three new developments. First, we introduce a modified energy objective, and hence modified Lagrangian, that permits both more transparent interpretation and more ready generalization. Second, we demonstrate that the bound gap  the difference between the upper and lower bounds for the desired output  can be represented as the sum of positive contributions  local indicators  associated with the elements TH of TH . Third, based on these local boundgap error indicators, we develop adaptive strategies by which to reduce the bound gap  and hence improve our validated prediction for the output of interest  through optimal refinement of TH . The resulting method is applied to an illustrative problem in linear elasticity.
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 33 (8 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.
Reference Jacobian OptimizationBased Rezone Strategies for Arbitrary Lagrangian Eulerian Methods
 Journal of Computational Physics
"... The philosophy of the Arbitrary LagrangianEulerian (ALE) methodology for solving multidimensional uid ow problems is to move the computational grid, using the ow as a guide, to improve the accuracy and eciency of the simulation. A principal element of ALE is the rezone phase in which a "rezoned" gr ..."
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Cited by 29 (14 self)
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The philosophy of the Arbitrary LagrangianEulerian (ALE) methodology for solving multidimensional uid ow problems is to move the computational grid, using the ow as a guide, to improve the accuracy and eciency of the simulation. A principal element of ALE is the rezone phase in which a "rezoned" grid is created that is adapted to the uid motion. We will describe a general rezone strategy that ensures the continuing geometric quality of the computational grid, while keeping "rezoned" grid at each time step as close as possible to the Lagrangian grid. Although the methodology can be applied to more general grid types, here we restrict ourselves to logically rectangular grids in two dimensions. The rezoning phase consists of two components: a sequence of local optimizations followed by a single global optimization. The local optimization denes a "reference" Jacobians which incorporates our denition of mesh quality at each point of the grid. The set of reference Jacobians then is used in the global optimization. At each node we form a local patch from the adjacent cells of Lagrangian grid and construct a local realization of the Winslow smoothness functional on this patch. Minimization of this functional with respect to position of central node denes its 'virtual' location (the node is not actually moved at this stage). By connecting this virtuallymoved node to its (stationary) neighbors, we dene a reference Jacobian that represents the best locally achievable geometric grid quality. The "rezoned" grid results from a minimization (where the points are actually moved) of a global objective function that measures the distance (in a leastsquares sense) between the Jacobian of the rezoned grid and the reference Jacobian. This objective function includes a "barrier" ...