MS-Mo-E-22
Recent Advances in A Posteriori Error Estimation and Adaptive Methods - Part II of II
For Part I, see MS-Mo-D-22

Date: August 10
Time: 16:00--18:30
Room: 206A

(Note: Click title to show the abstract.)

Organizer:
Zhang, Shun (City Univ. of Hong Kong)
Cai, Zhiqiang (Purdue Univ.)

Abstract: Self-adaptive numerical methods provide a powerful and automatic approach in scientific computing. In particular, Adaptive Mesh Refinement (AMR) algorithms have been widely used in computational science and engineering and have become a necessary tool in computer simulations of complex natural and engineering problems. The key ingredient for success of self-adaptive numerical methods is a posteriori error estimates that are able to accurately locate sources of global and local error in the current approximation. Talks in this mini-symposium will cover some recent advances in the development and analysis of both a posteriori estimators and (convergent) adaptive schemes, as well as indicate directions of future research.

MS-Mo-E-22-1
16:00--16:30
Robust a posteriori error analytical techniques for second and fourth order elliptic singularly perturbed problems
Shaohong, Du (Chongqing Jiaotong Univ./Beijing Computaional Sci. Research Center)
Zhimin, Zhang (Beijing Computational Sci. Research Center, & Wayne State Univ.)
Abstract: For singularly perturbed problems, a novel dual norm is introduced, under which robust residual-type and recovery-type a posteriori error estimators are analyzed. For fourth order elliptic singularly perturbed problems, a new size of the error for its mixed finite element methods is presented , and a novel analytical technique is developed to obtain robust residual-based a posteriori estimator in this size. Numerical experiments are reported to support theoretical results.

MS-Mo-E-22-2
16:30--17:00
Functional a posteriori error estimates and incompletely known data
Mali, Olli (Univ. of Jyväskylä)
Repin, Sergey (Univ. of Jyvaskyla)
Neittaanmäki, Pekka (Univ. of Jyväskylä)
Abstract: In this talk, the error estimates of functional type are used to study the effects of incompletely known data in problems generated by elliptic PDEs. PDEs typically contain parameters, which are known only up to some limited accuracy. The error estimates of functional type have suitable properties for a worst case scenario type analysis; they do not rely on any numerical method, they are guaranteed, and they depend explicitly on the problem data.

MS-Mo-E-22-3
17:00--17:30
Local H(div) recovery-based a posteriori error estimators for elliptic equations
Zhang, Xu (Purdue Univ.)
Cai, Zhiqiang (Purdue Univ.)
Abstract: We present recovery-based a posteriori error estimators for finite element approximation of elliptic equations. The flux is recovered in H(div) finite element subspaces by approximating equilibrium and constitutive equations simultaneously in a weighted H(div) norm. The recovery techniques are performed locally on appropriate patches of triangular elements. A posteriori error estimators are constructed based on difference of numerical flux and recovered flux. We will discuss the reliability and efficiency bounds of these local error estimators.

MS-Mo-E-22-4
17:30--18:00
Testing computationally the instance optimality of adaptive finite element methods
Veeser, Andreas (Univ. of Milan)
Fierro, Francesca (Univ. of Milan)
Schmidt, Alfred (Univ. of Bremen, Centre for Industrial Mathematics)
Abstract: Basing upon a localization of the error of the Ritz projection (Veeser, submitted) and adaptive tree approximation (Binev/DeVore '04), we approximately compute best errors with respect to the number of degrees of freedom and compare them with the corresponding errors of the adaptive finite element method. Our computational results complement the theoretical ones (e.g., Diening/Kreuzer/Stevenson '14) in which the error is augmented by the oscillation of the estimator.

MS-Mo-E-22-5
18:00--18:30
Robust adaptive approximation for singularly perturbed problems
Tantardini, Francesca (Ruhr-Universität Bochum)
Veeser, Andreas (Univ. of Milan)
Verfuerth, Ruediger (Ruhr-Universität Bochum)
Abstract: We consider the approximation in the reaction-diffusion norm with continuous finite elements and prove that the best error is equivalent to a sum of local best errors on pairs of elements. The equivalence constants do not depend on the ratio between reaction and diffusion. This allows in particular to derive local error functionals that ensure robust performance of adaptive tree approximation, which is useful e.g. to set non-asymptotic benchmarks for the corresponding adaptive Galerkin method.

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