A Posteriori Error Estimates of Residual Type for Second Order Quasi-Linear Elliptic PDEs (original) (raw)
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Optimal error properties of finite element methods for second order elliptic Dirichlet problems
Mathematics of Computation, 1982
We use the informational approach of Traub and Wozniakowski to study the variational form of the second order elliptic Dirichlet problem Lu = f on ü C RN. For /e Hr(Q), where r> -1, a quasi-uniform finite element method using n linear functional Jaf^i nas T7'(ß)-norm error 0(n~<r+1)/'v). We prove that it is asymptotically optimal among all methods using any information consisting of any n linear functionals. An analogous result holds if L is of order 2m: if / € Hr(ü), where r 3» -m, then there is a finite element method whose //"(fi)-norm error is %(n~(2m+r~a)/N) for 0 « a « m, and this is asymptotically optimal; thus, the optimal error improves as m increases. If the integrals jaf'p, are approximated by using n evaluations off, then there is a finite element method with quadrature with 7i'(ß)-norm error 0(n~r/N) where r > N/2. We show that when N = 1, there is no method using n function evaluations whose error is better than ñ(n~r); thus for N = 1, the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of /.
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Advances in Computational Mathematics, 2001
Techniques are developed for a posteriori error analysis of the non-homogeneous Dirichlet problem for the Laplacian giving computable error bounds for the error measured in the energy norm. The techniques are based on the equilibrated residual method that has proved to be reliable and accurate for the treatment of problems with homogeneous Dirichlet data. It is shown how the equilibrated residual method must be modified to include the practically important case of non-homogeneous Dirichlet data. Explicit and implicit a posteriori error estimators are derived and shown to be efficient and reliable. Numerical examples are provided illustrating the theory.
Error Estimates for the Finite Element Solutions of
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For plecewise linear approximation of variational inequalities asso- ciated with the mildly nonlinear elliptic boundary value problems having auxiliary constraint conditions, we prove that the error estimate for u-u h in the W 1'2norm is of order h. KEV WORDS AND PHRASES. Fine Element, V)nal Inequalities, Approximation, Mdly nonlinear. 1980 THEMATICS SUBJECT CLASSIFICATION CODES. Primary 5J20, 65N0, 41A15. In this paper, we derive the finite element error estimates for the approx- imate solution of mildly nonlinear boundary value problems having auxiliary con- straint conditions. A much used approach with any elliptic problem is to reform- ulate it in a weak for variational form It has been shown by Noor and Whlteman
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Thai Journal of Mathematics, 2015
We analyze a standard adaptive finite element method (AFEM) for second order semi-linear elliptic partial differential equations (PDEs) with vanishing boundary over a polyhedral domain in R d , d ≥ 2. Based on a posteriori error estimates using standard residual technique, we prove the contraction property for the weighted sum of the energy error and the error estimator between two consecutive iterations, which also leads to the convergence of AFEM. The obtained result is based on the assumptions that the initial mesh or triangulation is sufficiently refined and the nonlinear inhomogeneous term f (x, u(x)) is Lipschitz in the second variable.
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Abstract—We are concerned with an a posteriori error analysis of adaptive finite element approximations of boundary control problems for second order elliptic boundary value problems under bilateral bound constraints on the control which acts through a Neumann type boundary condition. In particular, the analysis of the errors in the state, the co-state, the control, and the co-control invokes an efficient and reliable residual-type a posteriori error estimator as well as data oscillations.
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