Issei Oikawa

Issei Oikawa

We propose and analyze a new hybridizable discontinuous Galerkin (HDG) method for second-order elliptic problems. Our method is obtained by inserting the $L^2$-orthogonal projection onto the approximate space for a numerical trace into all facet integrals in the usual HDG formulation. The orders of convergence for all variables are optimal if we use polynomials of degree $k+l$, $k+1$ and $k$, where $k$ and $l$ are any non-negative integers, to approximate the vector, scalar and trace variables, which implies that our method can achieve superconvergence for the scalar variable without postprocessing. Numerical results are presented to verify the theoretical results.

Takahito Kashiwabara, Issei Oikawa, Guanyu Zhou

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain $Ω\subset \mathbb R^N \, (N=2,3)$. 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 $u \cdot n_{\partialΩ} = g$ on $\partialΩ$. Because the original domain $Ω$ must be approximated by a polygonal (or polyhedral) domain $Ω_h$ before applying the finite element method, we need to take into account the errors owing to the discrepancy $Ω\neq Ω_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partialΩ}$ by $n_{\partialΩ_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator $H^1(Ω)^N \to H^{1/2}(\partialΩ)$; $u \mapsto u\cdot n_{\partialΩ}$. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates $O(h^α+ ε)$ and $O(h^{2α} + ε)$ for the velocity in the $H^1$- and $L^2$-norms respectively, where $α= 1$ if $N=2$ and $α= 1/2$ 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 $ε$ in the estimates.

Issei Oikawa

In this paper, we analyze a hybridized discontinuous Galerkin(HDG) method with reduced stabilization for the Stokes equations. The reduced stabilization enables us to reduce the number of facet unknowns and improve the computational efficiency of the method. We provide optimal error estimates in an energy and $L^2$ norms. It is shown that the reduced method with the lowest-order approximation is closely related to the nonconforming Crouzeix-Raviart finite element method. We also prove that the solution of the reduced method converges to the nonconforming Gauss-Legendre finite element solution as a stabilization parameter $τ$ tends to infinity and that the convergence rate is $O(τ^{-1})$.

Takahito Kashiwabara, Issei Oikawa, Guanyu Zhou

We consider the P1/P1 or P1b/P1 finite element approximations to the Stokes equations in a bounded smooth domain subject to the slip boundary condition. A penalty method is applied to address the essential boundary condition $u\cdot n = g$ on $\partialΩ$, which avoids a variational crime and simultaneously facilitates the numerical implementation. We give $O(h^{1/2} + ε^{1/2} + h/ε^{1/2})$-error estimate for velocity and pressure in the energy norm, where $h$ and $ε$ denote the discretization parameter and the penalty parameter, respectively. In the two-dimensional case, it is improved to $O(h + ε^{1/2} + h^2/ε^{1/2})$ by applying reduced-order numerical integration to the penalty term. The theoretical results are confirmed by numerical experiments.

Issei Oikawa

In this paper, we propose a hybridized discontinuous Galerkin(HDG) method with reduced stabilization for the Poisson equation. The reduce stabilization proposed here enables us to use piecewise polynomials of degree $k$ and $k-1$ for the approximations of element and inter-element unknowns, respectively, unlike the standard HDG methods. We provide the error estimates in the energy and $L^2$ norms under the chunkiness condition. In the case of $k=1$, it can be shown that the proposed method is closely related to the Crouzeix-Raviart nonconforming finite element method. Numerical results are presented to verify the validity of the proposed method.