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We extend the applicability of the popular interior penalty discontinuous Galerkin 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 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.

Background Esophageal squamous cell carcinoma (ESCC) is the leading cause of cancer death in men and women worldwide. The poor prognosis and rapid increase in ESCC incidence highlight the need to promote early detection and prediction. Identifying key molecular targets involved in ESCC monitoring and progression is critical for ESCC patients. Methods This study examined miR-25/miR-203 as a biomarker for ESCC patients. Real-time quantitative polymerase chain reaction (PCR) was used to detect miR-25/miR-203 expression levels in tissues and serum exosomes, and MiR-25/miR-203 upregulation was confirmed in ESCC. Results We found that the miR-25/miR-203 ratio in cancer tissues from 36 ESCC patients was significantly enhanced compared with that in adjacent tissues. Moreover, the serum level of miR-25/miR-203 in 57 ESCC patients was higher than that in 31 healthy volunteers. Intriguingly, in 38 ESCC patients, the level of miR-25/miR-203 decreased significantly after surgery. Using ROC curve statistical analysis, we found that each group of miR-25/miR-203 had obvious sensitivity and high specificity. The miR-25/miR-203 relationship with the clinicopathological features of ESCC patients was also analyzed. MiR-25/miR-203 was significantly associated with the ESCC TNM-stage and lymph node metastasis, which predicts the prognosis of ESCC and reflects tumor progression. Conclusions This study highlights the feasibility of using exosome-mediated miR-25/miR-203 as a vital noninvasive biomarker for the detection and treatment monitoring of ESCC.

International audience; We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C1-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H2 space and a generalised Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

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.

International audience; Discontinuous Galerkin (dG) methods on meshes consisting of polygonal/polyhedral (henceforth, collectively termed as polytopic) elements have received considerable attention in recent years. Due to the physical frame basis functions used typically and the quadrature challenges involved, the matrix-assembly step for these methods is often computationally cumbersome. To address this important practical issue, this work proposes two parallel assembly implementation algorithms on CUDA-enabled graphics cards for the interior penalty dG method on polytopic meshes for various classes of linear PDE problems. We are concerned with both single GPU parallelization, as well as with implementation on distributed GPU nodes. The results included showcase almost linear scalability of the quadrature step with respect to the number of GPU-cores used since no communication is needed for the assembly step. In turn, this can justify the claim that polytopic dG methods can be implemented extremely efficiently, as any assembly computing time overhead compared to finite elements on ‘standard’ simplicial or box-type meshes can be effectively circumvented by the proposed algorithms.

This chapter is devoted to studying the stability and convergence properties of the discontinuous Galerkin finite element method (DGFEM) for the numerical approximation of a general class of second-order partial differential equations (PDEs) with nonnegative characteristic form. While this class of second-order equations naturally includes parabolic equations, we also pursue the analysis of parabolic time-dependent PDEs in a separate manner in order to admit the use of local time-stepping algorithms. The general analysis of DGFEMs for PDEs with nonnegative characteristic form is pursued under the assumption that the number of faces each element possesses remains bounded under mesh refinement. For the special subclass of parabolic problems we adopt the assumption which permits an arbitrary number of faces per element, thereby highlighting that both assumptions lead to rigorous a priori error estimates for DGFEMs applied to parabolic problems.

This chapter is devoted to the automatic adaptive mesh refinement of polytopic meshes generated based on exploiting agglomeration of a given background geometry-conforming fine mesh. In particular, we exploit mesh partitioning techniques to design the underlying coarse mesh as well as for subdividing agglomerated elements marked for refinement. Here, the underlying adaptive refinement algorithm is constructed by employing goal-oriented dual-weighted-residual a posteriori error estimation techniques; in this setting, the aim of the computation is to accurately approximate the value of a given target or output functional of the solution. Numerical examples are presented to highlight the potential benefits of this approach when approximating partial differential equations posed on complicated domains.

In this chapter we study the stability and hp-version a priori error analysis of the discontinuous Galerkin finite element discretization of a pure diffusion problem. In particular, we develop the underlying theory for two different sets of shape assumptions which the polytopic elements forming the computational mesh must satisfy. In the first instance, we assume that the number of faces each element possesses remains uniformly bounded under mesh refinement, but without a restriction concerning shape-regularity. Secondly, we pursue the analysis in the case when this assumption is violated, i.e., when polytopic elements are permitted to have an arbitrary number of faces under mesh refinement; however, in this setting, a generalized shape-regularity assumption must be satisfied. The relationship between these different mesh assumptions is discussed in detail; indeed, the combination of these conditions allows for very general polytopic meshes to be admitted within our analysis.

This chapter develops the key mathematical tools needed to study the stability and convergence properties of hp-version discontinuous Galerkin finite element methods on polytopic meshes. A key issue in this setting is that general shape-regular polytopic meshes in \(\mathbb{R}^{d}\), d > 1, may, under mesh refinement, possess elements with (d − k)-dimensional facets, k = 1, 2, …, d − 1, which degenerate as the mesh size tends to zero. Thereby, care must be taken to ensure that the resulting inverse estimates and polynomial approximation results are sensitive to this type of degeneracy. The key approach adopted here is to exploit known results for standard elements, both within an L2- and L ∞ -setting, and to take the minimum of the resulting bounds. In this way, bounds which are optimal in both the h-version and p-version setting may be deduced, which directly account for (d − k)-dimensional facet degeneration, k = 1, 2, …, d − 1.

The purpose of this chapter is to present an overview of the construction of discontinuous Galerkin finite element methods for a general class of second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic partial differential equations, ultra-parabolic equations, first-order hyperbolic problems, the Kolmogorov-Fokker-Planck equations of Brownian motion, the equations of boundary layer theory in hydrodynamics, and various other degenerate equations.

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 hp-version 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 (P_p-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 P_p-basis on both polytopic and tensor-product elements with a (standard) DGFEM employing a (mapped) Q_p-basis. Moreover, a computational example is also presented which demonstrates the performance of the proposed hp-version DGFEM on general agglomerated meshes.

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.