244 research outputs found
Penalty method with Crouzeix-Raviart approximation for the Stokes equations under slip boundary condition
The Stokes equations subject to non-homogeneous slip boundary conditions are
considered in a smooth domain . We
propose a finite element scheme based on the nonconforming P1/P0 approximation
(Crouzeix-Raviart approximation) combined with a penalty formulation and with
reduced-order numerical integration in order to address the essential boundary
condition on . Because the
original domain must be approximated by a polygonal (or polyhedral)
domain before applying the finite element method, we need to take
into account the errors owing to the discrepancy , that
is, the issues of domain perturbation. In particular, the approximation of
by makes it non-trivial whether we
have a discrete counterpart of a lifting theorem, i.e., right-continuous
inverse of the normal trace operator ; . In this paper
we indeed prove such a discrete lifting theorem, taking advantage of the
nonconforming approximation, and consequently we establish the error estimates
and for the velocity in
the - and -norms respectively, where if and
if . This improves the previous result [T. Kashiwabara et
al., Numer. Math. 134 (2016), pp. 705--740] obtained for the conforming
approximation in the sense that there appears no reciprocal of the penalty
parameter in the estimates.Comment: 21 page
P1-Nonconforming finite elements on triangulations into triangles and quadrilaterals
The P1-nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected Lipschitz domains. An explicit a priori analysis for the combination of the Park–Sheen and the Crouzeix–Raviart nonconforming finite element methods is given for second-order elliptic PDEs with inhomogeneous Dirichlet boundary conditions
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
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