807 research outputs found
Convergence and Optimality of Adaptive Mixed Methods on Surfaces
In a 1988 article, Dziuk introduced a nodal finite element method for the
Laplace-Beltrami equation on 2-surfaces approximated by a piecewise-linear
triangulation, initiating a line of research into surface finite element
methods (SFEM). Demlow and Dziuk built on the original results, introducing an
adaptive method for problems on 2-surfaces, and Demlow later extended the a
priori theory to 3-surfaces and higher order elements. In a separate line of
research, the Finite Element Exterior Calculus (FEEC) framework has been
developed over the last decade by Arnold, Falk and Winther and others as a way
to exploit the observation that mixed variational problems can be posed on a
Hilbert complex, and Galerkin-type mixed methods can be obtained by solving
finite dimensional subproblems. In 2011, Holst and Stern merged these two lines
of research by developing a framework for variational crimes in abstract
Hilbert complexes, allowing for application of the FEEC framework to problems
that violate the subcomplex assumption of Arnold, Falk and Winther. When
applied to Euclidean hypersurfaces, this new framework recovers the original a
priori results and extends the theory to problems posed on surfaces of
arbitrary dimensions. In yet another seemingly distinct line of research,
Holst, Mihalik and Szypowski developed a convergence theory for a specific
class of adaptive problems in the FEEC framework. Here, we bring these ideas
together, showing convergence and optimality of an adaptive finite element
method for the mixed formulation of the Hodge Laplacian on hypersurfaces.Comment: 22 pages, no figures. arXiv admin note: substantial text overlap with
arXiv:1306.188
Convergence and Optimality of Adaptive Mixed Finite Element Methods
The convergence and optimality of adaptive mixed finite element methods for
the Poisson equation are established in this paper. The main difficulty for
mixed finite element methods is the lack of minimization principle and thus the
failure of orthogonality. A quasi-orthogonality property is proved using the
fact that the error is orthogonal to the divergence free subspace, while the
part of the error that is not divergence free can be bounded by the data
oscillation using a discrete stability result. This discrete stability result
is also used to get a localized discrete upper bound which is crucial for the
proof of the optimality of the adaptive approximation
Semilinear mixed problems on Hilbert complexes and their numerical approximation
Arnold, Falk, and Winther recently showed [Bull. Amer. Math. Soc. 47 (2010),
281-354] that linear, mixed variational problems, and their numerical
approximation by mixed finite element methods, can be studied using the
powerful, abstract language of Hilbert complexes. In another recent article
[arXiv:1005.4455], we extended the Arnold-Falk-Winther framework by analyzing
variational crimes (a la Strang) on Hilbert complexes. In particular, this gave
a treatment of finite element exterior calculus on manifolds, generalizing
techniques from surface finite element methods and recovering earlier a priori
estimates for the Laplace-Beltrami operator on 2- and 3-surfaces, due to Dziuk
[Lecture Notes in Math., vol. 1357 (1988), 142-155] and later Demlow [SIAM J.
Numer. Anal., 47 (2009), 805-827], as special cases. In the present article, we
extend the Hilbert complex framework in a second distinct direction: to the
study of semilinear mixed problems. We do this, first, by introducing an
operator-theoretic reformulation of the linear mixed problem, so that the
semilinear problem can be expressed as an abstract Hammerstein equation. This
allows us to obtain, for semilinear problems, a priori solution estimates and
error estimates that reduce to the Arnold-Falk-Winther results in the linear
case. We also consider the impact of variational crimes, extending the results
of our previous article to these semilinear problems. As an immediate
application, this new framework allows for mixed finite element methods to be
applied to semilinear problems on surfaces.Comment: 22 pages; v2: major revision, particularly sharpening of error
estimates in Section
Convergence of adaptive mixed finite element method for convection-diffusion-reaction equations
We prove the convergence of an adaptive mixed finite element method (AMFEM)
for (nonsymmetric) convection-diffusion-reaction equations. The convergence
result holds from the cases where convection or reaction is not present to
convection-or reaction-dominated problems. A novel technique of analysis is
developed without any quasi orthogonality for stress and displacement
variables, and without marking the oscillation dependent on discrete solutions
and data. We show that AMFEM is a contraction of the error of the stress and
displacement variables plus some quantity. Numerical experiments confirm the
theoretical results.Comment: arXiv admin note: text overlap with arXiv:1312.645
Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems
We prove that for compactly perturbed elliptic problems, where the
corresponding bilinear form satisfies a Garding inequality, adaptive
mesh-refinement is capable of overcoming the preasymptotic behavior and
eventually leads to convergence with optimal algebraic rates. As an important
consequence of our analysis, one does not have to deal with the a-priori
assumption that the underlying meshes are sufficiently fine. Hence, the overall
conclusion of our results is that adaptivity has stabilizing effects and can
overcome possibly pessimistic restrictions on the meshes. In particular, our
analysis covers adaptive mesh-refinement for the finite element discretization
of the Helmholtz equation from where our interest originated
- …