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.

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

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.

We extend the applicability of the popular interior-penalty discontinuous Galerkin (dG) method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace and Markov-type inverse estimates to arbitrary element shapes. A further new H 1 − L 2 -type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach.

We devise and analyze two hybrid high-order (HHO) methods for the numerical approximation of the biharmonic problem. The methods support polyhedral meshes, rely on the primal formulation of the problem, and deliver O(h k+1 ) H 2 -error estimates when using polynomials of order k ≥ 0 to approximate the normal derivative on the mesh (inter)faces. Both HHO methods hinge on a stabilization in the spirit of Lehrenfeld-Schöberl for second-order PDEs. The cell unknowns are polynomials of order (k + 2) that can be eliminated locally by means of static condensation. The face unknowns approximating the trace of the solution on the mesh (inter)faces are polynomials of order (k + 1) in the first HHO method which is valid in dimension two and uses an original stabilization involving the canonical hybrid finite element, and they are of order (k+2) for the second HHO method which is valid in arbitrary dimension and uses only L 2 -orthogonal projections in the stabilization. A comparative discussion with the weak Galerkin methods from the literature is provided, highlighting the close connections and the improvements proposed herein. Additionally, we show how the two HHO methods can be combined with a Nitsche-like boundary-penalty technique to weakly enforce the boundary conditions. An originality in the devised Nitsche's technique is to avoid any penalty parameter that must be large enough. Finally, numerical results showcase the efficiency of the proposed methods, and indicate that the HHO methods can generally outperform discontinuous Galerkin methods and even be competitive with C 0 -interior penalty methods on triangular meshes.

Recovered Finite Element Methods (R-FEM) have been recently introduced in Georgoulis and Pryer [Comput. Methods Appl. Mech. Eng. 332 (2018) 303–324]. for meshes consisting of simplicial and/or box-type elements. Here, utilising the flexibility of the R-FEM framework, we extend their definition to polygonal and polyhedral meshes in two and three spatial dimensions, respectively. An attractive feature of this framework is its ability to produce arbitrary order polynomial conforming discretizations, yet involving only as many degrees of freedom as discontinuous Galerkin methods over general polygonal/polyhedral meshes with potentially many faces per element. A priori error bounds are shown for general linear, possibly degenerate, second order advection-diffusion-reaction boundary value problems. A series of numerical experiments highlight the good practical performance of the proposed numerical framework.

BackgroundExosomes are nano-sized extracellular vesicles containing different biomolecules such as proteins and microRNAs (miRNAs) that mediate intercellular communication. Recently, numerous studies have reported the important functions of exosomal miRNAs in disease development and the potential clinical application as diagnostic biomarkers. Up to now, the most commonly used methods to extract exosomes are ultracentrifugation (UC) and precipitation-based commercial kit (e.g., ExoQuick). Generally, both UC and ExoQuick method could co-isolate contaminating proteins along with exosomes, with the UC method yielding even purer exosomes than ExoQuick. However, the comparison of these two methods on co-precipitated free miRNAs is still unknown.ResultsIn this study, we isolated exosomes from the human serum with exogenously added cel-miR-39 by UC and ExoQuick and compared the proportion of cel-miR-39 co-precipitated with exosomes extracted by these two methods. Using exogenous cel-miR-39 as free miRNAs in serum, we concluded that ExoQuick co-isolates a small proportion of free miRNAs while UC hardly precipitates any free miRNAs. We also found that incubation at 37℃ for 1 h could decrease the proportion of free miRNAs, and exosomal miRNAs like miR-126 and miR-152 also decreased when RNase A was used.ConclusionsIn conclusion, our findings provide essential information about the details of serum exosome isolation methods for further research on exosomal miRNAs.