1,175 research outputs found
Optimal Error Estimates for the hp–Version Interior Penalty Discontinuous Galerkin Finite Element Method
We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for second-order linear reaction-diffusion equations. To the best of our knowledge, the sharpest known error bounds for the hp-DGFEM are due to Riviere, Wheeler and Girault [9] and due to Houston, Schwab and Süli [6] which are optimal with respect to the meshsize h but suboptimal with respect to the polynomial degree p by half an order of p. We present improved error bounds in the energy norm, by introducing a new function space framework. More specifically, assuming that the solutions belong element-wise to an augmented Sobolev space, we deduce hp-optimal error bounds
A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems
We consider a variant of the hp-version interior penalty discontinuous Galerkin finite element method (IP-DGFEM) for second order problems of degenerate type. We do not assume uniform ellipticity of the diffusion tensor. Moreover, diffusion tensors or arbitrary form are covered in the theory presented. A new, refined recipe for the choice of the discontinuity-penalisation parameter (that is present in the formlation of the IP-DGFEM) is given. Making use of the recently introduced augmented Sobolev space framework, we prove an hp-optimal error bound in the energy norm and an h-optimal and slightly p-suboptimal (by only half an order of p) bound in the L2 norm, provided that the solution belongs to an augmented Sobolev space
Simulating Flaring Events in Complex Active Regions Driven by Observed Magnetograms
We interpret solar flares as events originating from active regions that have
reached the Self Organized Critical state, by using a refined Cellular
Automaton model with initial conditions derived from observations. Aims: We
investigate whether the system, with its imposed physical elements,reaches a
Self Organized Critical state and whether well-known statistical properties of
flares, such as scaling laws observed in the distribution functions of
characteristic parameters, are reproduced after this state has been reached.
Results: Our results show that Self Organized Criticality is indeed reached
when applying specific loading and relaxation rules. Power law indices obtained
from the distribution functions of the modeled flaring events are in good
agreement with observations. Single power laws (peak and total flare energy) as
well as power laws with exponential cutoff and double power laws (flare
duration) are obtained. The results are also compared with observational X-ray
data from GOES satellite for our active-region sample. Conclusions: We conclude
that well-known statistical properties of flares are reproduced after the
system has reached Self Organized Criticality. A significant enhancement of our
refined Cellular Automaton model is that it commences the simulation from
observed vector magnetograms, thus facilitating energy calculation in physical
units. The model described in this study remains consistent with fundamental
physical requirements, and imposes physically meaningful driving and
redistribution rules.Comment: 14 pages; 12 figures; 6 tables - A&A, in pres
hp-DGFEM on Shape-Irregular Meshes: Reaction-Diffusion Problems
We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order elliptic reaction-diffusion equations with mixed Dirichlet and Neumann boundary conditions. For simplicity of the presentation, we only consider boundary-value problems defined on an axiparallel polygonal domain whose solutions are approximated on subdivisions consisting of axiparallel elements. Our main concern is the generalisation of the error analysis of the hp-DGFEM for the case when shape-irregular (anisotropic) meshes and anisotropic polynomial degrees for the element basis functions are used. We shall present a general framework for deriving error bounds for the approximation error and we shall consider two important special cases. In the first of these we derive an error bound that is simultaneously optimal in h and p, for shape-regular elements and isotropic polynomial degrees, provided that the solution belongs to a certain anisotropic Sobolev space. The second result deals with the case where we have a uniform polynomial degree in every space direction and a shape-irregular mesh. Again we derive an error bound that is optimal both in h and in p. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement, in both cases considered. Finally, numerical experiments using shape-regular and shape-irregular elements are presented
Adaptive discontinuous Galerkin approximations to fourth order parabolic problems
An adaptive algorithm, based on residual type a posteriori indicators of
errors measured in and norms, for a numerical
scheme consisting of implicit Euler method in time and discontinuous Galerkin
method in space for linear parabolic fourth order problems is presented. The a
posteriori analysis is performed for convex domains in two and three space
dimensions for local spatial polynomial degrees . The a posteriori
estimates are then used within an adaptive algorithm, highlighting their
relevance in practical computations, which results into substantial reduction
of computational effort
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