229 research outputs found
On the threshold condition for Dörfler marking
It is an open question if the threshold condition θ θ_* the algebraic converges rate can be made arbitrarily small
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Adaptive Algorithms
Overwhelming empirical evidence in computational science and engineering proved that self-adaptive mesh-generation is a must-do in real-life problem computational partial differential equations. The mathematical understanding of corresponding algorithms concerns the overlap of two traditional mathematical disciplines, numerical analysis and approximation theory, with computational sciences. The half workshop was devoted to the mathematics of optimal convergence rates and instance optimality of the Dörfler marking or the maximum strategy in various versions of space discretisations and time-evolution problems with all kind of applications in the efficient numerical treatment of partial differential equations
Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity
In this work we propose and analyze a numerical method for electrical
impedance tomography of recovering a piecewise constant conductivity from
boundary voltage measurements. It is based on standard Tikhonov regularization
with a Modica-Mortola penalty functional and adaptive mesh refinement using
suitable a posteriori error estimators of residual type that involve the state,
adjoint and variational inequality in the necessary optimality condition and a
separate marking strategy. We prove the convergence of the adaptive algorithm
in the following sense: the sequence of discrete solutions contains a
subsequence convergent to a solution of the continuous necessary optimality
system. Several numerical examples are presented to illustrate the convergence
behavior of the algorithm.Comment: 26 pages, 12 figure
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Computational Engineering
The focus of this Computational Engineering Workshop was on the mathematical foundation of state-of-the-art and emerging finite element methods in engineering analysis. The 52 participants included mathematicians and engineers with shared interest on discontinuous Galerkin or Petrov-Galerkin methods and other generalized nonconforming or mixed finite element methods
An adaptive finite element scheme for the Hellinger--Reissner elasticity mixed eigenvalue problem
In this paper we study the approximation of eigenvalues arising from the
mixed Hellinger--Reissner elasticity problem by using the simple finite element
using partial relaxation of vertex continuity of stresses introduced
recently by Jun Hu and Rui Ma. We prove that the method converge when a
residual type error estimator is considered and that the estimator decays
optimally with respect to the number of degrees of freedom
The Prager-Synge theorem in reconstruction based a posteriori error estimation
In this paper we review the hypercircle method of Prager and Synge. This
theory inspired several studies and induced an active research in the area of a
posteriori error analysis. In particular, we review the Braess--Sch\"oberl
error estimator in the context of the Poisson problem. We discuss adaptive
finite element schemes based on two variants of the estimator and we prove the
convergence and optimality of the resulting algorithms
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