Unified analysis of HDG methods using scalar and vector hybrid variables

In this paper, hybridizable discontinuous Galerkin (HDG) methods using scalar and vector hybrid variables for steady-state diffusion problems are considered. We propose a unified framework to analyze the methods, where both the hybrid variables are treated as double-valued functions. If either of them is single valued, the well-posedness is ensured under some assumptions on approximation spaces. Moreover, we prove that all methods are superconvergent, based on the so-called $M$-decomposition theory. Numerical results are presented to validate our theoretical results.Comment: 16 page

Final Goods Substitutability and Economic Growth

In this paper, I investigate the effect of substitutability among final goods on welfare growth under the environment that productivity growth in each industry is not independent of one another. In such an environment, less substitutability is favorable to the welfare growth rate and the steady state welfare level, contrasting to Baumol (1967) and Lucas (1988).

Analysis of the Size of the Carcinoembryonic Antigen (CEA) Gene Family

Five members of the human CEA gene family [human pregnancy-specific β1-glycoprotein (PSβG), hsCGM1, 2, 3 and 4] have been isolated and identified through sequencing the exons containing their N-terminal domains. Sequence comparisons with published data for CEA and related molecules reveal the existence of highly-conserved gene subgroups within the CEA family. Together with published data eleven CEA family members have so far been determined. Apart from the highly conserved coding sequences, these genes also show strong sequence conservation in their introns, indicating a duplication of whole gene units during the evolution of the CEA gene family

Penalty method with Crouzeix-Raviart approximation for the Stokes equations under slip boundary condition

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain $\Omega \subset \mathbb R^N \, (N=2,3)$. We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix-Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition $u \cdot n_{\partial\Omega} = g$ on $\partial\Omega$. Because the original domain $\Omega$ must be approximated by a polygonal (or polyhedral) domain $\Omega_h$ before applying the finite element method, we need to take into account the errors owing to the discrepancy $\Omega \neq \Omega_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partial\Omega}$ by $n_{\partial\Omega_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator $H^1(\Omega)^N \to H^{1/2}(\partial\Omega)$; $u \mapsto u\cdot n_{\partial\Omega}$. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates $O(h^\alpha + \epsilon)$ and $O(h^{2\alpha} + \epsilon)$ for the velocity in the $H^1$- and $L^2$-norms respectively, where $\alpha = 1$ if $N=2$ and $\alpha = 1/2$ if $N=3$. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016), pp. 705--740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter $\epsilon$ in the estimates.Comment: 21 page