1,839 research outputs found
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
Convergence and optimality of the adaptive Morley element method
This paper is devoted to the convergence and optimality analysis of the
adaptive Morley element method for the fourth order elliptic problem. A new
technique is developed to establish a quasi-orthogonality which is crucial for
the convergence analysis of the adaptive nonconforming method. By introducing a
new parameter-dependent error estimator and further establishing a discrete
reliability property, sharp convergence and optimality estimates are then fully
proved for the fourth order elliptic problem
Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem
In this paper, we analyze the convergence and optimality of a standard
adaptive nonconforming linear element method for the Stokes problem. After
establishing a special quasi--orthogonality property for both the velocity and
the pressure in this saddle point problem, we introduce a new prolongation
operator to carry through the discrete reliability analysis for the error
estimator. We then use a specially defined interpolation operator to prove
that, up to oscillation, the error can be bounded by the approximation error
within a properly defined nonlinear approximate class. Finally, by introducing
a new parameter-dependent error estimator, we prove the convergence and
optimality estimates
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 Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods
In this article, we prove convergence of the weakly penalized adaptive
discontinuous Galerkin methods. Unlike other works, we derive the contraction
property for various discontinuous Galerkin methods only assuming the
stabilizing parameters are large enough to stabilize the method. A central idea
in the analysis is to construct an auxiliary solution from the discontinuous
Galerkin solution by a simple post processing. Based on the auxiliary solution,
we define the adaptive algorithm which guides to the convergence of adaptive
discontinuous Galerkin methods
The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements
For both the Poisson model problem and the Stokes problem in any dimension,
this paper proves that the enriched Crouzeix-Raviart elements are actually
identical to the first order Raviart-Thomas elements in the sense that they
produce the same discrete stresses. This result improves the previous result in
literature which, for two dimensions, states that the piecewise constant
projection of the stress by the first order Raviart-Thomas element is equal to
that by the Crouzeix-Raviart element. For the eigenvalue problem of Laplace
operator, this paper proves that the error of the enriched Crouzeix-Raviart
element is equivalent to that of the Raviart-Thomas element up to higher order
terms
Convergence of an adaptive mixed finite element method for general second order linear elliptic problems
The convergence of an adaptive mixed finite element method for general second
order linear elliptic problems defined on simply connected bounded polygonal
domains is analyzed in this paper. The main difficulties in the analysis are
posed by the non-symmetric and indefinite form of the problem along with the
lack of the orthogonality property in mixed finite element methods. The
important tools in the analysis are a posteriori error estimators,
quasi-orthogonality property and quasi-discrete reliability established using
representation formula for the lowest-order Raviart-Thomas solution in terms of
the Crouzeix-Raviart solution of the problem. An adaptive marking in each step
for the local refinement is based on the edge residual and volume residual
terms of the a posteriori estimator. Numerical experiments confirm the
theoretical analysis.Comment: 24 pages, 8 figure
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