38 research outputs found
Superconvergence of discontinuous Petrov-Galerkin approximations in linear elasticity
Existing a priori convergence results of the discontinuous Petrov-Galerkin
method to solve the problem of linear elasticity are improved. Using duality
arguments, we show that higher convergence rates for the displacement can be
obtained. Post-processing techniques are introduced in order to prove
superconvergence and numerical experiments {\color{black} confirm} our theory
Weakly symmetric stress equilibration for hyperelastic materialmodels
A stress equilibration procedure for hyperelastic material models is proposed
andanalyzed in this paper. Based on the displacement-pressure approximation
computed with a stable finite element pair, it constructs, in a
vertex-patch-wise manner, an -conforming approximation to the first
Piola-Kirchhoff stress. This is done in such a way that its associated Cauchy
stress is weakly symmetric in the sense that its anti-symmetric part is zero
tested against continuous piecewise linear functions. Our main result is the
identification of the subspace of test functions perpendicular to the range of
the local equilibration system on each patch which turn out to be rigid body
modes associated with the current configuration. Momentum balance properties
are investigated analytically and numerically and the resulting stress
reconstruction is shown to provide improved results for surface traction forces
by computational experiments
Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity
A stress equilibration procedure for linear elasticity is proposed and
analyzed in this paper with emphasis on the behavior for (nearly)
incompressible materials. Based on the displacement-pressure approximation
computed with a stable finite element pair, it constructs an -conforming, weakly symmetric stress reconstruction. Our focus is
on the Taylor-Hood combination of continuous finite element spaces of
polynomial degrees and for the displacement and the pressure,
respectively. Our construction leads then to reconstructed stresses by
Raviart-Thomas elements of degree which are weakly symmetric in the sense
that its anti-symmetric part is zero tested against continuous piecewise
polynomial functions of degree . The computation is performed locally on a
set of vertex patches covering the computational domain in the spirit of
equilibration \cite{BraSch:08}. Due to the weak symmetry constraint, the local
problems need to satisfy consistency conditions associated with all rigid body
modes, in contrast to the case of Poisson's equation where only the constant
modes are involved. The resulting error estimator is shown to constitute a
guaranteed upper bound for the error with a constant that depends only on the
shape regularity of the triangulation. Local efficiency, uniformly in the
incompressible limit, is deduced from the upper bound by the residual error
estimator
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
Convergence analysis of the scaled boundary finite element method for the Laplace equation
The scaled boundary finite element method (SBFEM) is a relatively recent
boundary element method that allows the approximation of solutions to PDEs
without the need of a fundamental solution. A theoretical framework for the
convergence analysis of SBFEM is proposed here. This is achieved by defining a
space of semi-discrete functions and constructing an interpolation operator
onto this space. We prove error estimates for this interpolation operator and
show that optimal convergence to the solution can be obtained in SBFEM. These
theoretical results are backed by a numerical example.Comment: 15 pages, 3 figure