Results 1  10
of
29
A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations
 475 (1994) MR 94j:65136
"... Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for spacetime finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizat ..."
Abstract

Cited by 70 (2 self)
 Add to MetaCart
Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for spacetime finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the socalled θscheme, which includes the implicit and explicit Euler methods and the CrankNicholson scheme. 1.
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 ..."
Abstract

Cited by 64 (9 self)
 Add to MetaCart
(Show Context)
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...
A Posteriori Error Control In LowOrder Finite Element Discretisations Of Incompressible Stationary Flow Problems
 Math. Comput
, 1999
"... . Computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and nonconforming finite element methods. A refined residualbased error estimate generalises th ..."
Abstract

Cited by 20 (7 self)
 Add to MetaCart
. Computable a posteriori error bounds and related adaptive meshrefining algorithms are provided for the numerical treatment of monotone stationary flow problems with a quite general class of conforming and nonconforming finite element methods. A refined residualbased error estimate generalises the works of Verfurth, Dari, Duran & Padra, Bao & Barrett. As a consequence, reliable and efficient averaging estimates can be established on unstructured grids. The symmetric formulation of the incompressible flow problem models certain nonNewtonian flow problems and the Stokes problem with mixed boundary conditions. A Helmholtz decomposition avoids any regularity or saturation assumption in the mathematical error analysis. Numerical experiments for the partly nonconforming method analysed by Kouhia and Stenberg indicate efficiency of related adaptive meshrefining algorithms. 1. Introduction Adaptive finite element methods play an important practical role in computational fluid dynamics. T...
Removing The Saturation Assumption In A Posteriori Error Analysis
"... this paper we prove the equivalence between BW and the true error, for piecewise linear finite elements, under minimal regularity of the data and no regularity of the continuous solution. We thus conclude that the saturation assumption is superfluous. The proof extends to the energy norm the pointwi ..."
Abstract

Cited by 13 (2 self)
 Add to MetaCart
this paper we prove the equivalence between BW and the true error, for piecewise linear finite elements, under minimal regularity of the data and no regularity of the continuous solution. We thus conclude that the saturation assumption is superfluous. The proof extends to the energy norm the pointwise argument discussed in [9].
A posteriori error estimates for the Stokes equations: a comparison
 Comput. Methods Appl. Mech. Engrg
, 1990
"... Several a posteriori error estimates for the Stokes equations have been derived by several authors. In this paper we compare some estimates based on the solution of local Stokes systems with estimates based on the residuals of the discretized finite element equations. Their performance as local indi ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
Several a posteriori error estimates for the Stokes equations have been derived by several authors. In this paper we compare some estimates based on the solution of local Stokes systems with estimates based on the residuals of the discretized finite element equations. Their performance as local indicators as well as global estimates is investigated. I.
The Efficient Location of Neighbors for Locally Refined nSimplicial Grids
 In 5th Int. Meshing Roundable
, 1996
"... . Grids of nsimplices frequently form a key ingredient when problems are modeled mathematically. In two and three space dimensions, such grids of triangles (respectively tetrahedrons) are used for the finite element approximation of solutions of partial differential equations (PDES), as in Kikuchi ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
. Grids of nsimplices frequently form a key ingredient when problems are modeled mathematically. In two and three space dimensions, such grids of triangles (respectively tetrahedrons) are used for the finite element approximation of solutions of partial differential equations (PDES), as in Kikuchi and Oden [24]. In more dimensions, nsimplices are used to approximate solution manifolds of parametrized equations (see Allgower and Georg [2] and Rheinboldt [39]) as well as for fixed point approximations of functions (see Todd [48]). Some types of simplicial methods may also efficiently replace quadtree and octtant representations of graphical data for post processing purposes, or for fractal image compression techniques, as shown in Hebert [23]. Finally, nvolume subdivision techniques are used for numerical integration purposes, as shown in Georg and Widmann [22], Allgower et al. [3] and Zumbusch in [49]. This paper presents a highly efficient bisection technique suited for the creation...
A Simple Analysis of Some A Posteriori Error Estimates
, 1997
"... Setting and Assumptions We consider the nonselfadjoint and possibly indefinite problem: find u 2 H such that B(u; v) = f(v) 8v 2 K; (2) where H and K are appropriate Hilbert spaces, B(\Delta; \Delta) is a bilinear form and f(\Delta) is a linear functional. With respect to the space H, we define a ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
Setting and Assumptions We consider the nonselfadjoint and possibly indefinite problem: find u 2 H such that B(u; v) = f(v) 8v 2 K; (2) where H and K are appropriate Hilbert spaces, B(\Delta; \Delta) is a bilinear form and f(\Delta) is a linear functional. With respect to the space H, we define an energy norm jjj \Delta jjj H associated with the positive definite scalar product (\Delta; \Delta) H jjjujjj 2 H = (u; u) H : (3) 2 With respect to the space K, we also define an energy norm jjj \Delta jjj K associated with the positive definite scalar product (\Delta; \Delta) K in a similar fashion. In order to insure that (2) has a unique solution, we assume the bilinear form B(\Delta; \Delta) satisfies the continuity condition jB(OE; j)j jjjOEjjj H jjjjjjj K 8OE 2 H; 8j 2 K: (4) We also assume the infsup conditions inf OE 2 H jjjOEjjj H = 1 sup j 2 K jjjjjjj K 1 B(OE; j) ? 0; (5) sup OE2H B(OE; j) ? 0; j 6= 0; j 2 K: (6) Let M h ae H and N h ae K be members of two fam...
Computational survey on a posteriori error estimators for nonconforming finite element methods for Poisson problems
 J. Comput. Appl. Math
, 2013
"... Abstract. This survey compares different strategies for guaranteed error control for the lowestorder nonconforming CrouzeixRaviart finite element method for the Stokes equations. The upper error bound involves the minimal distance of the computed piecewise gradient DNC uCR to the gradients of Sob ..."
Abstract

Cited by 4 (4 self)
 Add to MetaCart
Abstract. This survey compares different strategies for guaranteed error control for the lowestorder nonconforming CrouzeixRaviart finite element method for the Stokes equations. The upper error bound involves the minimal distance of the computed piecewise gradient DNC uCR to the gradients of Sobolev functions with exact boundary conditions. Several improved suggestions for the cheap computation of such test functions compete in five benchmark examples. This paper provides numerical evidence that guaranteed error control of the nonconforming FEM is indeed possible for the Stokes equations with overall effectivity indices between 1 to 4. 1.
A posteriori dualmixed adaptive finite element error control for Lamé and
 Stokes equations, Numer. Math
"... Abstract A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the HellingerReissner principle. This quasioptimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) i ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract A unified and robust mathematical model for compressible and incompressible linear elasticity can be obtained by rephrasing the Herrmann formulation within the HellingerReissner principle. This quasioptimally converging extension of PEERS (Plane Elasticity Element with Reduced Symmetry) is called DualMixed Hybrid formulation (DMH). Explicit residualbased a posteriori error estimates for DMH are introduced and are mathematically shown to be lockingfree, reliable, and efficient. The estimator serves as a refinement indicator in an adaptive algorithm for effective automatic mesh generation. Numerical evidence supports that the adaptive scheme leads to optimal convergence for Lamé and Stokes benchmark problems with singularities. 1 Introduction and