Takahito Kashiwabara

Takahito Kashiwabara

A finite element approximation of the Stokes equations under a certain nonlinear boundary condition, namely, the slip or leak boundary condition of friction type, is considered. We propose an approximate problem formulated by a variational inequality, prove an existence and uniqueness result, present an error estimate, and discuss a numerical realization using an iterative Uzawa-type method. Several numerical examples are provided to support our theoretical results.

Takahito Kashiwabara, Tomoya Kemmochi

Pointwise error analysis of the linear finite element approximation for $-Δu + u = f$ in $Ω$, $\partial_n u = τ$ on $\partialΩ$, where $Ω$ is a bounded smooth domain in $\mathbb R^N$, is presented. We establish $O(h^2|\log h|)$ and $O(h)$ error bounds in the $L^\infty$- and $W^{1,\infty}$-norms respectively, by adopting the technique of regularized Green's functions combined with local $H^1$- and $L^2$-estimates in dyadic annuli. Since the computational domain $Ω_h$ is only polyhedral, one has to take into account non-conformity of the approximation caused by the discrepancy $Ω_h \neq Ω$. In particular, the so-called Galerkin orthogonality relation, utilized three times in the proof, does not exactly hold and involves domain perturbation terms (or boundary-skin terms), which need to be addressed carefully. A numerical example is provided to confirm the theoretical result.

Takahito Kashiwabara, Tomoya Kemmochi

In this paper, we consider the finite element approximation for a parabolic problem on a smooth domain $Ω\subset \mathbb{R}^N$ with the inhomogeneous Neumann boundary condition. We emphasize that the domain can be non-convex in general. We implement the finite element method for this problem by constructing a family of polygonal or polyhedral domains $\{ Ω_h \}_h$ that approximate the original domain $Ω$. The main result of this study is the $L^\infty$-error estimate for this approximation. We shall show that the convergence rate is not optimal for higher order elements since the symmetric difference $Ω\bigtriangleup Ω_h$ is not empty in general. In order to address the effect of the symmetric difference of domains, we introduce the tubular neighborhood of the original boundary $\partialΩ$. We will also present a slightly new approach to establish the $L^\infty$-error estimate. Moreover, we present the smoothing property for the discrete parabolic semigroup and the spatially discretized maximal regularity as corollaries of the main result.

Daisuke Inoue, Yuji Ito, Takahito Kashiwabara et al.

When controlling multi-agent systems, the trade-off between performance and scalability is a major challenge. Here, we address this difficulty by using mean field games (MFGs), which is a framework that deduces the macroscopic dynamics describing the density profile of agents from their microscopic dynamics. To effectively use the MFG, we propose a model predictive MFG (MP-MFG), which estimates the agent population density profile with using kernel density estimation and manages the input generation with model predictive control. The proposed MP-MFG generates control inputs by monitoring the agent population at each time step, and thus achieves higher robustness than the conventional MFG. Numerical results show that the MP-MFG outperforms the MFG when the agent model has modeling errors or the number of agents in the system is small.

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.

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.