205 research outputs found
A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem
AbstractIn this work we derive and analyze a posteriori error estimators for low-order nonconforming finite element methods of the linear elasticity problem on both triangular and quadrilateral meshes, with hanging nodes allowed for local mesh refinement. First, it is shown that equilibrated Neumann data on interelement boundaries are simply given by the local weak residuals of the numerical solution. The first error estimator is then obtained by applying the equilibrated residual method with this set of Neumann data. From this implicit estimator we also derive two explicit error estimators, one of which is similar to the one proposed by Dörfler and Ainsworth (2005) [24] for the Stokes problem. It is established that all these error estimators are reliable and efficient in a robust way with respect to the Lamé constants. The main advantage of our error estimators is that they yield guaranteed, i.e., constant-free upper bounds for the energy-like error (up to higher order terms due to data oscillation) when a good estimate for the inf–sup constant is available, which is confirmed by some numerical results
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
Quasi-optimal adaptive hybridized mixed finite element methods for linear elasticity
For the planar Navier--Lam\'e equation in mixed form with symmetric stress
tensors, we prove the uniform quasi-optimal convergence of an adaptive method
based on the hybridized mixed finite element proposed in [Gong, Wu, and Xu:
Numer.~Math., 141 (2019), pp.~569--604]. The main ingredients in the analysis
consist of a discrete a posteriori upper bound and a quasi-orthogonality result
for the stress field under the mixed boundary condition. Compared with existing
adaptive methods, the proposed adaptive algorithm could be directly applied to
the traction boundary condition and be easily implemented
A Locking-Free Weak Galerkin Finite Element Method for Linear Elasticity Problems
In this paper, we introduce and analyze a lowest-order locking-free weak
Galerkin (WG) finite element scheme for the grad-div formulation of linear
elasticity problems. The scheme uses linear functions in the interior of mesh
elements and constants on edges (2D) or faces (3D), respectively, to
approximate the displacement. An -conforming displacement
reconstruction operator is employed to modify test functions in the right-hand
side of the discrete form, in order to eliminate the dependence of the
parameter in error estimates, i.e., making the scheme
locking-free. The method works without requiring to be bounded. We prove optimal error estimates, independent of
, in both the -norm and the -norm. Numerical experiments
validate that the method is effective and locking-free
Nonstandard Finite Element Methods
[no abstract available
Taylor-Hood like finite elements for nearly incompressible strain gradient elasticity problems
We propose a family of mixed finite elements that are robust for the nearly
incompressible strain gradient model, which is a fourth-order singular
perturbed elliptic system. The element is similar to [C. Taylor and P. Hood,
Comput. & Fluids, 1(1973), 73-100] in the Stokes flow. Using a uniform discrete
B-B inequality for the mixed finite element pairs, we show the optimal rate of
convergence that is robust in the incompressible limit. By a new regularity
result that is uniform in both the materials parameter and the
incompressibility, we prove the method converges with order to the
solution with strong boundary layer effects. Moreover, we estimate the
convergence rate of the numerical solution to the unperturbed second-order
elliptic system. Numerical results for both smooth solutions and the solutions
with sharp layers confirm the theoretical prediction.Comment: 27 pages, 1 figures, 4 table
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