MSMoE23
Recent Developments in Finite Element Methods for
Variational Inequalities  Part II of II
For Part I, see MSMoD23
Date: August 10
Time: 16:0018:10
Room: 208A
(Note: Click title to show the abstract.)
Organizer:
Nataraj, Neela (Indian Inst. of Tech. Bombay)
Gudi, Thirupathi (Indian Inst. of Sci., Bangalore)
Abstract: Variational inequalities have been playing a key role in
the modern scientific world. The theory of variational
inequalities provides a generalization of the theory of boundary
value problems and has applications in many fields like Applied
Mathematics, Mechanics, Theory of Control and so on. Unlike
variational equations, inequalities exhibit additional
singularities due to occurrence of free boundaries, which limit
the regularity of the solution. The study of computational methods
for variational inequalities thus offers more challenges. The
error analysis for the finite element methods of these problems
should also be derived under the limited regularity assumptions.
Adaptive finite element techniques are quite desirable for these
class of problems. We would like to discuss and exchange some of
the latest developments in the error analysis of finite element
methods for variational inequalities.
MSMoE231
16:0016:30
Lagrange multipliers in the a posteriori error analysis for obstacle problems
Veeser, Andreas (Univ. of Milan)
Abstract: Sharp a posteriori error estimators for obstacle problems rely on an approximation of the Lagrange multiplier associated with the exact solution. The construction of such approximate multipliers must consider structural conditions arising from the variational inequality, the associated potential and the request of computability. We shall discuss the interplay of tools like full contact (Fierro/Veeser '03) and positivity preserving interpolation (Chen/Nochetto '00, Nochetto/Wahlbin '01).
MSMoE232
16:3017:00
Adaptive nonconforming FEM for the Obstacle Problem
Koehler, Karoline (HumboldtUniversitaet zu Berlin)
Abstract: This talk considers the nonconforming CrouzeixRaviart finite element
method (NCFEM) for the discretisation of the obstacle problem. The
presented a priori error analysis employs the standard regularity
assumption and shows convergence even in the case of problems on
polygonal domains with reentering corners. The striking advantage of the
NCFEM is the possibility to compute lower bounds for the exact minimal
energy. This is a novel result and not possible with an conforming finite
element method.
MSMoE233
17:0017:30
Optimal Convergence Rates of Adaptive Simulations in Elastoplasticity
Carstensen, Carsten (HumboldtUniversitaet zu Berlin)
Abstract: An adaptive finite element algorithm for problems in elastoplasticity
with hardening is of optimal convergence with respect to
the notion of approximation classes. The results rely on the equivalence of the
errors of the stresses and energies resulting from Jensen's inequality. Numerical
experiments study the influence of the hardening and bulk parameters to
the convergence behavior of the AFEM algorithm. This is the first optimal
adaptive FEM for a variational inequality to appear in the online version
in Nummer. Math. (2015) with Andreas Schroeder and Sebastian Wiedemann.
CPMoE234
17:3017:50
Computable error estimates for Monte Carlo finite element approximation of elliptic PDE with lognormal diffusion coefficients
Hall, Eric (KTH Royal Inst. of Tech.)
Hoel, Haakon (Univ. of Oslo)
Sandberg, Mattias (KTH Royal Inst. of Tech.)
Szepessy, Anders (KTH Royal Inst. of Tech.)
TEMPONE, RAUL (KING ABDULLAH Univ. OF Sci. & Tech.)
Abstract: The Monte Carlo (and Multilevel Monte Carlo) finite element method can be used to approximate observables of solutions to diffusion equations with lognormal distributed diffusion coefficients, e.g. modeling ground water flow. Typical models use lognormal diffusion coefficients with Hoelder regularity of order up to 1/2 a.s. This low regularity implies that the high frequency finite element approximation error (i.e. the error from frequencies larger than the mesh frequency) is not negligible and can be larger than the computable low frequency error. We address how the total error can be estimated by the computable error.
CPMoE235
17:5018:10
Finite element analyses on optimal control problems constrained by Stochastic PDEs
Sun, Tongjun (School of Mathematics, Shandong Univ.)
Abstract: We consider optimal control problem governed by PDEs with stochastic perturbation in its coefficients. The objective is to minimize the expectation of a cost functional with the constrained control. We represent the stochastic PDEs in term of the generalized polynomial chaos expansion and obtain the deterministic optimal problem. By applying the wellknown Lions' Lemma, we obtain the necessary and sufficient optimality conditions. We establish a scheme to approximate the optimality system with respect to both the spatial space and the probability space by Stochastic Galerkin method. Then priori error estimates are derived for the state, the costate and the control variables. Numerical examples are presented to illustrate our theoretical results.
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Footnote: Code: TypeDateTimeRoom No.
Type : IL=Invited Lecture, SL=Special Lectures, MS=Minisymposia, IM=Industrial Minisymposia, CP=Contributed Papers, PP=Posters
Date: Mo=Monday, Tu=Tuesday, We=Wednesday, Th=Thursday, Fr=Friday
Time : A=8:309:30, B=10:0011:00, C=11:1012:10, BC=10:0012:10, D=13:3015:30, E=16:0018:00, F=19:0020:00, G=12:1013:30, H=15:3016:00
Room No.: TBA
