Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analysed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (Pp-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a Pp-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem.

We present in a unified framework new conforming and nonconforming Virtual Element Methods (VEM) for general second order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and non-symmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal H 1 -and L 2 -error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.

A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription.For more information, please contact eprints@nottingham.ac.uk Mathematical Modelling and Numerical AnalysisWill be set by the publisher Modélisation Mathématique et Analyse Numérique Abstract. We consider the hp-version interior penalty discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of the advection-diffusion-reaction equation on general computational meshes consisting of polygonal/polyhedral (polytopic) elements. In particular, new hpversion a priori error bounds are derived based on a specific choice of the interior penalty parameter which allows for edge/face-degeneration. The proposed method employs elemental polynomial bases of total degree p (Pp-basis) defined in the physical coordinate system, without requiring the mapping from a given reference or canonical frame. Numerical experiments highlighting the performance of the proposed DGFEM are presented. In particular, we study the competitiveness of the p-version DGFEM employing a Pp-basis on both polytopic and tensor-product elements with a (standard) DGFEM employing a (mapped) Qp-basis. Moreover, a computational example is also presented which demonstrates the performance of the proposed hp-version DGFEM on general agglomerated meshes. hp-VERSION DISCONTINUOUS GALERKIN METHODS FOR ADVECTION-DIFFUSION-REACTION PROBLEMS ON POLYTOPIC MESHES ANDREA CANGIANI

An posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing.

Abstract. We present a new hp-version space-time discontinuous Galerkin (dG) finite element method for the numerical approximation of parabolic evolution equations on general spatial meshes consisting of polygonal/polyhedral (polytopic) elements, giving rise to prismatic space-time elements. A key feature of the proposed method is the use of space-time elemental polynomial bases of total degree, say p, defined in the physical coordinate system, as opposed to standard dG-time-stepping methods whereby spatial elemental bases are tensorized with temporal basis functions. This approach leads to a fully discrete hp-dG scheme using fewer degrees of freedom for each time step, compared to dG time-stepping schemes employing tensorized space-time basis, with acceptable deterioration of the approximation properties. A second key feature of the new space-time dG method is the incorporation of very general spatial meshes consisting of possibly polygonal/polyhedral elements with arbitrary number of faces. A priori error bounds are shown for the proposed method in various norms. An extensive comparison among the new space-time dG method, the (standard) tensorized space-time dG methods, the classical dG-time-stepping, and conforming finite element method in space, is presented in a series of numerical experiments.Key words. space-time discontinuous Galerkin; hp-finite element methods; reduced cardinality basis functions; discontinuous Galerkin time-stepping.AMS subject classifications. 65N30, 65M60, 65J101. Introduction. The discontinuous Galerkin (dG) method can be traced back to [41], where it was introduced as a nonstandard finite element scheme for solving the neutron transport equation. This dG method was analyzed in [36], where it was also applied as a time stepping scheme for initial value problem for ordinary differential equations, and was shown to be equivalent to certain implicit Runge-Kutta methods. Jamet [34] introduced a dG time-stepping scheme for parabolic problems on evolving domains, later extended and analysed in [24,20,21,22,23]. For an introduction, we refer to the classic monograph [52] and the references therein. In [38], the quasioptimality of the dG time-stepping method for parabolic problems in mesh-dependent norms is established. Also, dG time-stepping convergence analyses under minimal regularity were shown in [57,14,15]. In all aforementioned literature, convergence of the discrete solution to the exact solution is achieved by reducing spatial mesh size h and time step size τ at some fixed (typically low) order.On the other hand, the p-and hp-version finite element method (FEM) appeared in the 1980s (see [7,6], and also the textbook [46] for a extensive survey). p-and hp-version FEM can achieve exponential rates of convergence when the underlying solution is locally analytic by increasing the polynomial order p and/or locally grading the meshsize towards corner or edge singularities. In this vein, the analyticity in the time-variable in parabolic problems has given rise to the use of p-and hp-versi...

The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements.

SUMMARYIn this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C 0 Lagrange finite element methods. In the virtual element method, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of C 0 -continuous virtual element methods on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation, and present a virtual element approach for the isoparametric four-node quadrilateral and the eight-node hexahedral element. In addition, we show quantitative * Correspondence to: N. Sukumar,

Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analysed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (Pp-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a Pp-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem.