In this paper we consider the a posteriori and a priori error analysis of discontinuous Galerkin interior penalty methods for second-order partial differential equations with nonnegative characteristic form on anisotropically refined computational meshes. In particular, we discuss the question of error estimation for linear target functionals, such as the outflow flux and the local average of the solution. Based on our a posteriori error bound we design and implement the corresponding adaptive algorithm to ensure reliable and efficient control of the error in the prescribed functional to within a given tolerance. This involves exploiting both local isotropic and anisotropic mesh refinement. The theoretical results are illustrated by a series of numerical experiments.

In this article I discuss Bernard Williams' realist conception of legitimacy. According to his critics Williams tacitly incorporates various moral claims, endorses a philosophically suspect 'consensus' view of politics, and employs an unrealistic and moralised conception of political rule. I argue that these criticisms mischaracterise the nature of the basic legitimation demand and the judgements about the acceptability of the state at its core and conclude that political theorists who object to the direction and style of much contemporary political theory should take seriously the possibility of developing an appropriately 'political' political theory on Williamsian lines.Keywords: Bernard Williams; basic legitimation demand; political realism; political moralism A number of thinkers concerned by the style of much contemporary political theory have recently attacked the suggestion that political theory is a branch of applied ethics and gestured towards the possibility of developing a more self-consciously realist approach. Chief among these is Bernard Williams, whose posthumous political essays have been influential in setting the agenda of the burgeoning realist movement in political theory. The basic legitimation demand (BLD) is central to Williams' realism because he claims it offers a distinctively political standard of evaluation that avoids the moralism of much contemporary political theory. However, commentators have recently disputed this claim by arguing that Williams tacitly incorporates moral claims, which precludes him from offering a realist alternative to political moralism, commits to a philosophically suspect consensus view of politics and endorses an unrealistic and moralised conception of politics itself. Hence, one critic concludes that Williams' realism 'shares significant features with liberal theory ... which means that it is vulnerable to the same critique that other realists have made of liberalism' (Sleat, 2010, p. 486) while another insists that it offers an 'unfeasible return to an ideal-type expectation' (Freeden, 2012, p. 6).In this article I focus on four influential criticisms because by engaging with Williams' critics we can better appreciate the attractions of thinking about politics in the terms he encourages. I argue that Williams' critics mischaracterise the nature of the BLD and the judgements about the acceptability of the state at its core, and conclude that political theorists who are uneasy with the moralism of much contemporary political theory should re-evaluate the attractions of thinking about politics in Williams' terms. The First Political QuestionWilliams claims that two forms of political moralism (PM) dominate political philosophy. Enactment models, like utilitarianism, formulate 'principles, concepts, ideals, and values' and seek to 'express these in political action', while structural models, like those of Rawls, spell bs_bs_banner

In recent years a number of realist thinkers have charged much contemporary political theory with being idealistic and moralistic. While the basic features of the realist counter-movement are reasonably well understood, realism is still considered a critical, primarily negative creed which fails to offer a positive, alternative way of thinking normatively about politics. Aiming to counteract this general perception, in this article I draw on Bernard Williams's claims about how to construct a politically coherent conception of liberty from the non-political value of freedom. I do this because Williams's argument provides an illuminating example of the distinctive nature of realist political thinking and its attractions. I argue that Williams's account of realist political thinking challenges the orthodox moralist claim that normative political arguments must be guided by an ideal ethical theory. I then spell out the repercussions Williams's claims about the significance of political opposition and non-moralised accounts of motivation have for our understanding of the role and purpose of political theory. I conclude by defending the realist claim that action-guiding political theory should accordingly take certain features of our politics as given, most centrally the reality of political opposition and the passions and experiences that motivate them. On this reading political realism offers a viable way of thinking about political values which cannot be understood in terms of the categories of intellectual separation -ideal/nonideal or factinsensitive/fact-sensitive -that have marked political theory in recent years.The jumble, the mayhem, the mess proved itself more subtle than this one's ideology and that one's morality.

In this paper we present a residual-based a posteriori error estimator for hp-adaptive discontinuous Galerkin methods for elliptic eigenvalue problems. In particular, we use as a model problem the Laplace eigenvalue problem on bounded domains in ℝd, d = 2, 3, with homogeneous Dirichlet boundary conditions. Analogous error estimators can be easily obtained for more complicated elliptic eigenvalue problems. We prove the reliability and efficiency of the residual-based error estimator also for non-convex domains and use numerical experiments to show that, under an hp-adaptation strategy driven by the error estimator, exponential convergence can be achieved, even for non-smooth eigenfunctions.

We consider the a priori error analysis of hp-version interior penalty discontinuous Galerkin methods for second-order partial differential equations with nonnegative characteristic form under weak assumptions on the mesh design and the local finite element spaces employed. In particular, we prove a priori hp-error bounds for linear target functionals of the solution, on (possibly) anisotropic computational meshes with anisotropic tensor-product polynomial basis functions. The theoretical results are illustrated by a numerical experiment.