53 research outputs found
On the convergence of adaptive nonconforming finite element methods for a class of convex variational problems
We formulate and analyze an adaptive nonconforming finite element method for the solution of convex variational problems. The class of minimization problems we admit includes highly singular problems for which no Euler–Lagrange equation (or inequality) is available. As a consequence, our arguments only use the structure of the energy functional. We are nevertheless able to prove convergence of an adaptive algorithm, using even refinement indicators that are not reliable error indicators
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 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
Convergent adaptive hybrid higher-order schemes for convex minimization
This paper proposes two convergent adaptive mesh-refining algorithms for the
hybrid high-order method in convex minimization problems with two-sided
p-growth. Examples include the p-Laplacian, an optimal design problem in
topology optimization, and the convexified double-well problem. The hybrid
high-order method utilizes a gradient reconstruction in the space of piecewise
Raviart-Thomas finite element functions without stabilization on triangulations
into simplices or in the space of piecewise polynomials with stabilization on
polytopal meshes. The main results imply the convergence of the energy and,
under further convexity properties, of the approximations of the primal resp.
dual variable. Numerical experiments illustrate an efficient approximation of
singular minimizers and improved convergence rates for higher polynomial
degrees. Computer simulations provide striking numerical evidence that an
adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even
with empirical higher convergence rates
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