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Open access publications by faculty, postdocs, and graduate students in the Department of Mathematical Sciences

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    Sequestration of gene products by decoys enhances precision in the timing of intracellular events
    (Scientific Reports, 2024-11-08) Biswas, Kuheli; Dey, Supravat; Singh, Abhyudai
    Expressed gene products often interact ubiquitously with binding sites at nucleic acids and macromolecular complexes, known as decoys. The binding of transcription factors (TFs) to decoys can be crucial in controlling the stochastic dynamics of gene expression. Here, we explore the impact of decoys on the timing of intracellular events, as captured by the time taken for the levels of a given TF to reach a critical threshold level, known as the first passage time (FPT). Although nonlinearity introduced by binding makes exact mathematical analysis challenging, employing suitable approximations and reformulating FPT in terms of an alternative variable, we analytically assess the impact of decoys. The stability of the decoy-bound TFs against degradation impacts FPT statistics crucially. Decoys reduce noise in FPT, and stable decoy-bound TFs offer greater timing precision with less expression cost than their unstable counterparts. Interestingly, when both bound and free TFs decay at the same rate, decoy binding does not directly alter FPT noise. We verify these results by performing exact stochastic simulations. These results have important implications for the precise temporal scheduling of events involved in the functioning of biomolecular clocks, development processes, cell-cycle control, and cell-size homeostasis.
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    Towards highly efficient thin-film solar cells with a graded-bandgap CZ TSSe layer. Part II: Piecewise-homogeneous bandgap grading
    (JPhys Energy, 2024-11-14) Ahmad, Faiz; Monk, Peter B.; Lakhtakia, Akhlesh
    In Part I, we optoelectronically optimized a thin-film solar cell with a graded-bandgap CZTSSe photon-absorbing layer and a periodically corrugated backreflector, using the hybridizable discontinuous Galerkin (HDG) scheme to solve the drift-diffusion equations. The efficiency increase due to periodic corrugation was minimal, but significant improvement was achieved with a nonlinearly graded bandgap. Due to occasional failures of the HDG scheme from negative carrier densities, we developed a new computational scheme using the finite-difference method, which also reduced the overall computational cost of optimization. Later, a normalization error was discovered in the electrical submodel in Part I, but it did not alter the overall conclusions. We have now re-optimized the solar cells with (i) a homogeneous bandgap, (ii) a linearly graded bandgap, or (iii) a nonlinearly graded bandgap, and (iv) a piecewise-homogeneous bandgap which is easier to implement than a continuously graded bandgap. An efficiency of 13.53% is predicted with a three-layered piecewise-homogeneous CZTSSe layer. Furthermore, concentrating sunlight by a factor of one hundred can boost the efficiency to 16.70% with the piecewise-homogeneous bandgap.
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    A Moments-Based Analytical Approach for Cell Size Homeostasis
    (IEEE Control Systems Letters, 2024-06-07) Nieto, César; Vargas-Garcia, Cesar Augusto; Singh, Abhyudai
    This contribution explores mechanisms that regulate the dynamics of single-cell size, maintaining equilibrium around a target set point. Using the formalism of Stochastic Hybrid Systems (SHS), we consider continuous exponential growth in cell size (as determined by volume/mass/surface area). This continuous-time evolution is interspersed by cell division events that occur randomly as per a given size-dependent rate, and upon division, only one of the two daughter cells is tracked. We show that a size-independent division rate does not provide cell size homeostasis, in the sense that the variance in cell size increases unboundedly over time. Next, we consider a division rate proportional to cell size that yields the adder size control observed in several bacteria – a constant size is added on average between birth and division regardless of the newborn size. For this scenario, we obtain exact formulas for the steady-state moments (mean, variance, and skewness) of cell size. Expanding the SHS model, we explore a biologically relevant scenario where the time between successive division events is further divided into multiple discrete stages with size-dependent stage transitions. Exact moment computations demonstrate that increasing the number of stages reduces cell size variability (noise). We also find formulas considering uneven size partitioning between daughters during division, and where the division rate follows a power law of the cell size leading to deviations from adder size control. This letter provides a method for estimating model parameters from observed cell size distributions and enhances our understanding of mechanisms underlying cell size regulation.
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    Teaching Mathematics through Problem Posing: four practices for handling students’ posed problems
    (Revemop, 2024-08-17) Ma, Yue; Cai, Jinfa
    Problem posing is both a learning goal and an effective instructional approach, but what are the details of using this approach to foster students’ learning? In this paper, we use teaching cases to demonstrate a step-by-step instructional model that teachers can use to convert non-problem-posing tasks into problem-posing ones and use the tasks to foster students’ learning. More importantly, we provide a detailed illustration of the four practices that teachers and students can use to handle students’ posed problems. The paper ends with a discussion of future work to validate and improve the model and the challenges that teachers experience teaching through problem posing. Resumen Plantear problemas es tanto un objetivo de aprendizaje como un enfoque didáctico eficaz, pero ¿cuáles son los detalles de la utilización de este enfoque para promover el aprendizaje de los alumnos? En este artículo, utilizamos casos de enseñanza para demostrar un modelo de instrucción paso a paso que los profesores pueden utilizar para convertir tareas que no son de resolución de problemas en tareas de resolución de problemas y utilizarlas para promover el aprendizaje de los alumnos. Y lo que es más importante, ofrecemos una ilustración detallada de las cuatro prácticas que profesores y alumnos pueden utilizar para tratar los problemas propuestos por los estudiantes. El artículo termina con una discusión sobre el trabajo futuro para validar y mejorar el modelo y los retos a los que se enfrentan los profesores cuando enseñan a través del planteamiento de problemas.
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    The Future of Mathematics Education Research: A Discussion Group
    (Proceedings of the 14th International Congress on Mathematical Education, 2024-06) Bakker, Arthur; Cai, Jinfa
    With help of a review study by Inglis and Foster published in Journal for Research in Mathematics Education, Jinfa Cai summarized trends in the past 50 years of mathematics education research. Next, Arthur Bakker presented a recent survey published in Educational Studies in Mathematics about the future of mathematics education research. Anna Sfard compared this survey with an earlier survey for ICMI. The presentations were discussed in the whole group, after which Jill Adler highlighted a few points she considered relevant.
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    A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids
    (Electronic Research Archive, 2024-05-27) Yang, Yan; Ye, Xiu; Zhang, Shangyou
    A pressure-robust stabilizer-free weak Galerkin (WG) finite element method has been defined for the Stokes equations on triangular and tetrahedral meshes. We have obtained pressure-independent error estimates for the velocity without any velocity reconstruction. The optimal-order convergence for the velocity of the WG approximation has been proved for the L2 norm and the H1 norm. The optimal-order error convergence has been proved for the pressure in the L2 norm. The theory has been validated by performing some numerical tests on triangular and tetrahedral meshes.
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    Graduate Teaching Assistants’ Perception of Student Difficulties and Use in Teaching
    (International Journal of Research in Undergraduate Mathematics Education, 2024-06-01) Park, Jungeun; Rizzolo, Douglas
    Given the important role graduate teaching assistants (TAs) play in undergraduate students’ learning, we investigated what TAs identified as students’ difficulties from students’ written work, their plans to address them, and implementation of their plans in class. Since the difficulties that TAs identified in general matched errors that students made, we analyzed what TAs identified in terms of literature on error handling. We examined levels of specific details of students’ work involved in TAs’ identifying, planning, and teaching. Our results show that (a) TAs often did not identify the most frequent errors students made, which reflected well-documented difficulties from the literature, (b) the errors TAs identified were mainly procedural in nature, (c) specific details of students’ work were mainly included in procedural errors, and (d) the level of specificity of students’ work was generally consistent but showed some drops when going from identifying to planning, then to teaching. Our results highlight interesting questions for future research and could be used as resources to design professional development that helps TAs use students’ errors in teaching to promote students’ learning.
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    On the analysis of modes in a closed electromagnetic waveguide
    (Comptes Rendus Mathématique, 2024-05-31) Halla, Martin; Monk, Peter
    Modal expansions are useful to understand wave propagation in an infinite closed electromagnetic waveguide. They can also be used to construct generalized Dirichlet-to-Neumann maps that provide artificial boundary conditions for truncating a computational domain when discretizing the field by finite elements. The modes of a waveguide arise as eigenfunctions of a non-symmetric eigenvalue problem, and the eigenvalues determine the propagation (or decay) of the modes along the waveguide. For the successful use of waveguide modes, it is necessary to know that the modes exist and form a dense set in a suitable function space containing the trace of the electric field in the waveguide. This paper is devoted to proving such a density result using the methods of Keldysh. We also show that the modes satisfy a useful orthogonality property, and show how the Dirichlet-to-Neumann map can be calculated. Our existence and density results are proved under realistic regularity assumptions on the cross section of the waveguide, and the electromagnetic properties of the materials in the waveguide, so generalizing existing results. Résumé Les expansions modales sont utiles pour comprendre la propagation des ondes dans un guide d’ondes électromagnétiques fermé et infini. Elles peuvent également être utilisées pour construire des cartes de Dirichlet à Neumann généralisées qui peuvent être utilisées pour fournir des conditions limites artificielles afin de tronquer un domaine de calcul lors de la discrétisation du champ par des éléments finis. Les modes d’un guide d’ondes apparaissent comme des fonctions propres d’un problème de valeurs propres non symétrique, et les valeurs propres déterminent la propagation (ou la décroissance) des modes le long du guide d’ondes. Pour une utilisation réussie des modes de guide d’ondes, il est nécessaire de savoir que les modes existent et forment un ensemble dense dans un espace de fonctions approprié contenant la trace du champ électrique dans le guide d’ondes. Cet article est consacré à la démonstration d’un tel résultat de densité en utilisant les méthodes de Keldysh. Nous montrons également que les modes satisfont une propriété d’orthogonalité utile, et nous montrons comment la carte de Dirichlet à Neumann peut être calculée. Nos résultats d’existence et de densité sont prouvés sous des hypothèses de régularité réalistes sur la section transversale du guide d’ondes et les propriétés électromagnétiques des matériaux dans le guide d’ondes, généralisant ainsi les résultats existants.
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    An Entropy Power Inequality for Dependent Variables
    (IEEE Transactions on Information Theory, 2024-04-05) Madiman, Mokshay; Melbourne, James; Roberto, Cyril
    The entropy power inequality for independent random variables is a foundational result of information theory, with deep connections to probability and geometric functional analysis. Very few extensions of the entropy power inequality have been developed for settings with dependence. We address this gap in the literature by developing entropy power inequalities for dependent random variables. In particular, we highlight the role of log-supermodularity in delivering sufficient conditions for an entropy power inequality stated using conditional entropies.
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    Bifacial flexible CIGS thin-film solar cells with nonlinearly graded-bandgap photon-absorbing layers
    (JPhys Energy, 2024-03-06) Ahmad, Faiz; Monk, Peter B.; Lakhtakia, Akhlesh
    The building sector accounts for 36% of energy consumption and 39% of energy-related greenhouse-gas emissions. Integrating bifacial photovoltaic solar cells in buildings could significantly reduce energy consumption and related greenhouse gas emissions. Bifacial solar cells should be flexible, bifacially balanced for electricity production, and perform reasonably well under weak-light conditions. Using rigorous optoelectronic simulation software and the differential evolution algorithm, we optimized symmetric/asymmetric bifacial CIGS solar cells with either (i) homogeneous or (ii) graded-bandgap photon-absorbing layers and a flexible central contact layer of aluminum-doped zinc oxide to harvest light outdoors as well as indoors. Indoor light was modeled as a fraction of the standard sunlight. Also, we computed the weak-light responses of the CIGS solar cells using LED illumination of different light intensities. The optimal bifacial CIGS solar cell with graded-bandgap photon-absorbing layers is predicted to perform with 18%–29% efficiency under 0.01–1.0-Sun illumination; furthermore, efficiencies of 26.08% and 28.30% under weak LED light illumination of 0.0964 mW cm−2 and 0.22 mW cm−2 intensities, respectively, are predicted.
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    Beyond computation: Assessing in-service mathematics teachers’ conceptual understanding of fraction division through problem posing
    (Asian Journal for Mathematics Education, 2023-12-15) Yao, Yiling; Jia, Suijun; Cai, Jinfa
    Problem posing has long been recognized as a critically important teaching method and goal in the area of mathematics education. However, few studies have used problem posing to assess in-service teachers’ mathematical understanding. The present study investigated in-service teachers’ mathematical understanding of fraction division, which is often considered challenging content in elementary school, from three angles: computation, drawing, and problem posing. Two studies involving 66 and 193 primary and middle school teachers were conducted to reveal the in-service teachers’ mathematical understanding and whether drawing and problem posing affected each other. Although the in-service teachers rarely had the opportunity to pose mathematical problems in their daily teaching, they were able to pose mathematical problems in this study. In addition, problem-posing tasks were more useful in diagnosing the in-service teachers’ conceptual understanding than were computation or drawing. Thus, problem posing seems to have contributed to their conceptual understanding of fraction division on the drawing task.
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    A superconvergent CDG finite element for the Poisson equation on polytopal meshes
    (Zeitschrift für anorganische und allgemeine Chemie | Journal of Inorganic and General Chemistry, 2023-12-08) Ye, Xiu; Zhang, Shangyou
    A conforming discontinuous Galerkin (CDG) finite element is constructed for solving second order elliptic equations on polygonal and polyhedral meshes. The numerical trace on the edge between two elements is no longer the average of two discontinuous Pk functions on the two sides, but a lifted Pk+2 function from four Pk functions. When the numerical gradient space is the H (div,T) subspace of piecewise Pdk+1 polynomials on subtriangles/subtehrahedra of a polygon/polyhedron T which have a one-piece polynomial divergence on T, this CDG method has a superconvergence of order two above the optimal order. Due to the superconvergence, we define a post-process which lifts a Pk CDG solution to a quasi-optimal Pk+2 solution on each element. Numerical examples in 2D and 3D are computed and the results confirm the theory.
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    Saddle point least squares discretization for convection-diffusion
    (Applicable Analysis, 2023-12-11) Bacuta, Constantin; Hayes, Daniel; O'Grady, Tyler
    We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models.
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    Fascination with Fluctuation: Luria and Delbrück’s Legacy
    (Axioms, 2023-03-07) Robeva, Raina S.; Jungck, John R.
    While Luria and Delbrück’s seminal work has found its way to some college biology textbooks, it is now largely absent from those in mathematics. This is a significant omission, and we consider it a missed opportunity to present a celebrated conceptual model that provides an authentic and, in many ways, intuitive example of the quantifiable nature of stochasticity. We argue that it is an important topic that could enrich the educational literature in mathematics, from the introductory to advanced levels, opening many doors to undergraduate research. The paper has two main parts. First, we present in detail the mathematical theory behind the Luria–Delbrück model and make suggestions for further readings from the literature. We also give ideas for inclusion in various mathematics courses and for projects that can be used in regular courses, independent projects, or as starting points for student research. Second, we briefly review available hands-on activities as pedagogical ways to facilitate problem posing, problem-based learning, and investigative case-based learning and to expose students to experiments leading to Poisson distributions. These help students with even limited mathematics backgrounds understand the significance of Luria–Delbrück’s work for determining mutation rates and its impact on many fields, including cancer chemotherapy, antibiotic resistance, radiation, and environmental screening for mutagens and teratogens.
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    Fundamental limits of parasitoid-driven host population suppression: Implications for biological control
    (PLoS ONE, 2023-12-22) Singh, Abhyudai
    Parasitoid wasps are increasingly being used to control insect pest populations, where the pest is the host species parasitized by the wasp. Here we use the discrete-time formalism of the Nicholson-Bailey model to investigate a fundamental question—are there limits to parasitoid-driven suppression of the host population density while still ensuring a stable coexistence of both species? Our model formulation imposes an intrinsic self-limitation in the host’s growth resulting in a carrying capacity in the absence of the parasitoid. Different versions of the model are considered with parasitism occurring at a developmental stage that is before, during, or after the growth-limiting stage. For example, the host’s growth limitation may occur at its larval stage due to intraspecific competition, while the wasps attack either the host egg, larval or pupal stage. For slow-growing hosts, models with parasitism occurring at different life stages are identical in terms of their host suppression dynamics but have contrasting differences for fast-growing hosts. In the latter case, our analysis reveals that wasp parasitism occurring after host growth limitation yields the lowest pest population density conditioned on stable host-parasitoid coexistence. For ecologically relevant parameter regimes we estimate this host suppression to be roughly 10-20% of the parasitoid-free carrying capacity. We further expand the models to consider a fraction of hosts protected from parasitism (i.e., a host refuge). Our results show that for a given host reproduction rate there exists a critical value of protected host fraction beyond which, the system dynamics are stable even for high levels of parasitism that drive the host to arbitrary low population densities. In summary, our systematic analysis sheds key insights into the combined effects of density-dependence in host growth and parasitism refuge in stabilizing the host-parasitoid population dynamics with important implications for biological control.
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    Multi-frequency Linear Sampling Method on Experimental Data Sets
    (IEEE Transactions on Antennas and Propagation, 2023-08-01) Monk, Peter; Pena, Manuel; Selgas, Virginia
    We investigate the use of the Linear Sampling Method (LSM) for determining the shape of a scatterer from multi-frequency experimental data. We study three multi-frequency indicators for two 2D data sets available online: one is provided by the Institut Fresnel, and another by the Electromagnetic Imaging Laboratory of the University of Manitoba. We show that the multi-frequency LSM works exceptionally well on the 2D Fresnel database, and also acceptably well on the Manitoba one. In particular, a new multi-frequency indicator is tested, and data completion for the Fresnel data set is studied. We also test an adaptive technique to cut down on the number of evaluations of the indicator function for well resolved scatterers.
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    Integral representation of hydraulic permeability
    (Proceedings of the Royal Society of Edinburgh Section A: Mathematics, 2022-05-06) Bi, Chuan; Ou, Miao-Jung Yvonne; Zhang, Shangyou
    In this paper, we show that the permeability of a porous material (Tartar (1980)) and that of a bubbly fluid (Lipton and Avellaneda. Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79) are limiting cases of the complexified version of the two-fluid models posed in Lipton and Avellaneda (Proc. R. Soc. Edinburgh Sect. A: Math. 114 (1990), 71–79). We assume the viscosity of the inclusion fluid is zμ1 and the viscosity of the hosting fluid is μ1∈R+ , z∈C . The proof is carried out by the construction of solutions for large |z| and small |z| with an iteration process similar to the one used in Bruno and Leo (Arch. Ration. Mech. Anal. 121 (1993), 303–338) and Golden and Papanicolaou (Commun. Math. Phys. 90 (1983), 473–491) and the analytic continuation. Moreover, we also show that for a fixed microstructure, the permeabilities of these three cases share the same integral representation formula (3.17) with different values of contrast parameter s:=1/(z−1) , as long as s is outside the interval [−2E221+2E22,−11+2E21] , where the positive constants E1 and E2 are the extension constants that depend only on the geometry of the periodic pore space of the material.
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    Effects of defect density, minority carrier lifetime, doping density, and absorber-layer thickness in CIGS and CZTSSe thin-film solar cells
    (Journal of Photonics for Energy, 2023-06-02) Ahmad, Faiz; Civiletti, Benjamin J.; Monk, Peter B.; Lakhtakia, Akhlesh
    Detailed optoelectronic simulations of thin-film photovoltaic solar cells (PVSCs) with a homogeneous photon-absorber layer made of with CIGS or CZTSSe were carried out to determine the effects of defect density, minority carrier lifetime, doping density, composition (i.e., bandgap energy), and absorber-layer thickness on solar-cell performance. The transfer-matrix method was used to calculate the electron-hole-pair (EHP) generation rate, and a one-dimensional drift-diffusion model was used to determine the EHP recombination rate, open-circuit voltage, short-circuit current density, power-conversion efficiency, and fill factor. Through a comparison of limited experimental data and simulation results, we formulated expressions for the defect density in terms of the composition parameter of either CIGS or CZTSSe. All performance parameters of the thin-films PVSCs were thereby shown to be obtainable from the bulk material-response parameters of the semiconductor, with the influence of surface defects being small enough to be ignored. Furthermore, unrealistic values of the defect density (equivalently, minority carrier lifetime) will deliver unreliable predictions of the solar-cell performance. The derived expressions should guide fellow researchers in simulating the graded-bandgap and quantum-well-based PVSCs.
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    Transcriptional regulation of Sis1 promotes fitness but not feedback in the heat shock response
    (eLife, 2023-05-17) Grade, Rania; Singh, Abhyudai; Ali, Asif; Pincus, David
    The heat shock response (HSR) controls expression of molecular chaperones to maintain protein homeostasis. Previously, we proposed a feedback loop model of the HSR in which heat-denatured proteins sequester the chaperone Hsp70 to activate the HSR, and subsequent induction of Hsp70 deactivates the HSR (Krakowiak et al., 2018; Zheng et al., 2016). However, recent work has implicated newly synthesized proteins (NSPs) – rather than unfolded mature proteins – and the Hsp70 co-chaperone Sis1 in HSR regulation, yet their contributions to HSR dynamics have not been determined. Here, we generate a new mathematical model that incorporates NSPs and Sis1 into the HSR activation mechanism, and we perform genetic decoupling and pulse-labeling experiments to demonstrate that Sis1 induction is dispensable for HSR deactivation. Rather than providing negative feedback to the HSR, transcriptional regulation of Sis1 by Hsf1 promotes fitness by coordinating stress granules and carbon metabolism. These results support an overall model in which NSPs signal the HSR by sequestering Sis1 and Hsp70, while induction of Hsp70 – but not Sis1 – attenuates the response.
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    Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes
    (CSIAM Transactions on Applied Mathematics, 2023-02) Ye, Xiu; Zhang, Shangyou
    It is known that discontinuous finite element methods use more unknown variables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation using discontinuous Pk elements on triangular and tetrahedral meshes. Our new CDG method maximizes the potential of discontinuous Pk element in order to improve the convergence rate. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and in the L2 norm. A local post-process is defined which lifts a Pk CDG solution to a discontinuous Pk+2 solution. It is proved that the lifted Pk+2 solution converges at the optimal order. The numerical tests confirm the theoretic findings. Numerical comparison is provided in 2D and 3D, showing the Pk CDG finite element is as good as the Pk+2 continuous Galerkin finite element.
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