Adaptive discontinuous Galerkin approximations to fourth order parabolic problems

An adaptive algorithm, based on residual type a posteriori indicators of errors measured in $L^{\infty}(L^2)$ and $L^2(L^2)$ 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 $r\ge 2$. 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

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

Validation and Benchmarking of a Practical Free Magnetic Energy and Relative Magnetic Helicity Budget Calculation in Solar Magnetic Structures

In earlier works we introduced and tested a nonlinear force-free (NLFF) method designed to self-consistently calculate the free magnetic energy and the relative magnetic helicity budgets of the corona of observed solar magnetic structures. The method requires, in principle, only a single, photospheric or low-chromospheric, vector magnetogram of a quiet-Sun patch or an active region and performs calculations in the absence of three-dimensional magnetic and velocity-field information. In this work we strictly validate this method using three-dimensional coronal magnetic fields. Benchmarking employs both synthetic, three-dimensional magnetohydrodynamic simulations and nonlinear force-free field extrapolations of the active-region solar corona. We find that our time-efficient NLFF method provides budgets that differ from those of more demanding semi-analytical methods by a factor of ~3, at most. This difference is expected from the physical concept and the construction of the method. Temporal correlations show more discrepancies that, however, are soundly improved for more complex, massive active regions, reaching correlation coefficients of the order of, or exceeding, 0.9. In conclusion, we argue that our NLFF method can be reliably used for a routine and fast calculation of free magnetic energy and relative magnetic helicity budgets in targeted parts of the solar magnetized corona. As explained here and in previous works, this is an asset that can lead to valuable insight into the physics and the triggering of solar eruptions.Comment: 32 pages, 14 figures, accepted by Solar Physic

Validation of the magnetic energy vs. helicity scaling in solar magnetic structures

We assess the validity of the free magnetic energy - relative magnetic helicity diagram for solar magnetic structures. We used two different methods of calculating the free magnetic energy and the relative magnetic helicity budgets: a classical, volume-calculation nonlinear force-free (NLFF) method applied to finite coronal magnetic structures and a surface-calculation NLFF derivation that relies on a single photospheric or chromospheric vector magnetogram. Both methods were applied to two different data sets, namely synthetic active-region cases obtained by three-dimensional magneto-hydrodynamic (MHD) simulations and observed active-region cases, which include both eruptive and noneruptive magnetic structures. The derived energy--helicity diagram shows a consistent monotonic scaling between relative helicity and free energy with a scaling index 0.84$\pm$0.05 for both data sets and calculation methods. It also confirms the segregation between noneruptive and eruptive active regions and the existence of thresholds in both free energy and relative helicity for active regions to enter eruptive territory. We consider the previously reported energy-helicity diagram of solar magnetic structures as adequately validated and envision a significant role of the uncovered scaling in future studies of solar magnetism

Energy and helicity budgets of solar quiet regions

We investigate the free magnetic energy and relative magnetic helicity budgets of solar quiet regions. Using a novel non-linear force-free method requiring single solar vector magnetograms we calculate the instantaneous free magnetic energy and relative magnetic helicity budgets in 55 quiet-Sun vector magnetograms. As in a previous work on active regions, we construct here for the first time the (free) energy-(relative) helicity diagram of quiet-Sun regions. We find that quiet-Sun regions have no dominant sense of helicity and show monotonic correlations a) between free magnetic energy/relative helicity and magnetic network area and, consequently, b) between free magnetic energy and helicity. Free magnetic energy budgets of quiet-Sun regions represent a rather continuous extension of respective active-region budgets towards lower values, but the corresponding helicity transition is discontinuous due to the incoherence of the helicity sense contrary to active regions. We further estimate the instantaneous free magnetic-energy and relative magnetic-helicity budgets of the entire quiet Sun, as well as the respective budgets over an entire solar cycle. Derived instantaneous free magnetic energy budgets and, to a lesser extent, relative magnetic helicity budgets over the entire quiet Sun are comparable to the respective budgets of a sizeable active region, while total budgets within a solar cycle are found higher than previously reported. Free-energy budgets are comparable to the energy needed to power fine-scale structures residing at the network, such as mottles and spicules

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