174 research outputs found
Quasi-optimality of a pressure-robust nonconforming finite element method for the Stokes problem
Nearly all classical inf-sup stable mixed finite element methods for the incompressible Stokes equations are not pressure-robust, i.e., the velocity error is dependent on the pressure. However, recent results show that pressure-robustness can be recovered by a non-standard discretization of the right hand side alone. This variational crime introduces a consistency error in the method which can be estimated in a straightforward manner provided that the exact velocity solution is sufficiently smooth. The purpose of this paper is to analyze the pressure-robust scheme with low regularity. The numerical analysis applies divergence-free H¹-conforming Stokes finite element methods as a theoretical tool. As an example, pressure-robust velocity and pressure a-priori error estimates will be presented for the (first order) nonconforming Crouzeix--Raviart element. A key feature in the analysis is the dependence of the errors on the Helmholtz projector of the right hand side data, and not on the entire data term. Numerical examples illustrate the theoretical results
Instance optimal Crouzeix-Raviart adaptive finite element methods for the Poisson and Stokes problems
We extend the ideas of Diening, Kreuzer, and Stevenson [Instance optimality
of the adaptive maximum strategy, Found. Comput. Math. (2015)], from conforming
approximations of the Poisson problem to nonconforming Crouzeix-Raviart
approximations of the Poisson and the Stokes problem in 2D. As a consequence,
we obtain instance optimality of an AFEM with a modified maximum marking
strategy
Convergence and optimality of an adaptive modified weak Galerkin finite element method
An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is
studied in this paper, in addition to its convergence and optimality. The weak
Galerkin bilinear form is simplified without the need of the skeletal variable,
and the approximation space is chosen as the discontinuous polynomial space as
in the discontinuous Galerkin method. Upon a reliable residual-based a
posteriori error estimator, an adaptive algorithm is proposed together with its
convergence and quasi-optimality proved for the lowest order case. The major
tool is to bridge the connection between weak Galerkin method and the
Crouzeix-Raviart nonconforming finite element. Unlike the traditional
convergence analysis for methods with a discontinuous polynomial approximation
space, the convergence of AmWG is penalty parameter free
Innovative Approaches to the Numerical Approximation of PDEs
This workshop was about the numerical solution of PDEs for which classical
approaches,
such as the finite element method, are not well suited or need further
(theoretical) underpinnings.
A prominent example of PDEs for which classical methods are not well
suited are PDEs posed in high space dimensions.
New results on low rank tensor approximation for those problems were
presented.
Other presentations dealt with regularity of PDEs, the numerical solution
of PDEs on surfaces,
PDEs of fractional order, numerical solvers for PDEs that converge with
exponential rates, and
the application of deep neural networks for solving PDEs
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