1,458 research outputs found
Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error
Compared to conforming P1 finite elements, nonconforming P1 finite element
discretizations are thought to be less sensitive to the appearance of distorted
triangulations. E.g., optimal-order discrete norm best approximation
error estimates for functions hold for arbitrary triangulations. However,
similar estimates for the error of the Galerkin projection for second-order
elliptic problems show a dependence on the maximum angle of all triangles in
the triangulation. We demonstrate on the example of a special family of
distorted triangulations that this dependence is essential, and due to the
deterioration of the consistency error. We also provide examples of sequences
of triangulations such that the nonconforming P1 Galerkin projections for a
Poisson problem with polynomial solution do not converge or converge at
arbitrarily slow speed. The results complement analogous findings for
conforming P1 elements.Comment: 23 pages, 10 figure
A Nonconforming Finite Element Approximation for the von Karman Equations
In this paper, a nonconforming finite element method has been proposed and
analyzed for the von Karman equations that describe bending of thin elastic
plates. Optimal order error estimates in broken energy and norms are
derived under minimal regularity assumptions. Numerical results that justify
the theoretical results are presented.Comment: The paper is submitted to an international journa
A posteriori error control for discontinuous Galerkin methods for parabolic problems
We derive energy-norm a posteriori error bounds for an Euler time-stepping
method combined with various spatial discontinuous Galerkin schemes for linear
parabolic problems. For accessibility, we address first the spatially
semidiscrete case, and then move to the fully discrete scheme by introducing
the implicit Euler time-stepping. All results are presented in an abstract
setting and then illustrated with particular applications. This enables the
error bounds to hold for a variety of discontinuous Galerkin methods, provided
that energy-norm a posteriori error bounds for the corresponding elliptic
problem are available. To illustrate the method, we apply it to the interior
penalty discontinuous Galerkin method, which requires the derivation of novel a
posteriori error bounds. For the analysis of the time-dependent problems we use
the elliptic reconstruction technique and we deal with the nonconforming part
of the error by deriving appropriate computable a posteriori bounds for it.Comment: 6 figure
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