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Hiroki Ishizaka

Publications and source records attributed to Hiroki Ishizaka.

At least 19 recordsLinked to original sources
arXiv2026

Anisotropic Lagrange interpolation on exact spherical triangles

We study linear Lagrange interpolation on exact geodesic spherical triangles obtained by radial projection of chordal triangles. Factorising the element map into an affine map and a radial correction separates chordal anisotropy from radial distortion. We derive exact identities for the surface measure, Dirichlet energy, and directional derivatives, and use them to transfer planar anisotropic interpolation estimates to the sphere. The resulting local error estimates retain the two chordal directional length scales and keep the geometric factors explicit, with constants independent of the element size, shape, and sphere radius. In the gradient estimate, the maximum-angle factor multiplies only the longer directional scale. Four explicit triangle families quantify the loss caused by replacing these scales with the diameter and establish the sharp maximum-angle dependence. They also show that, within the same two-term directional bound with a constant independent of the element, the maximum-angle factor cannot be moved to the shorter scale or have its exponent reduced below one. The fourth family proves the optimality of the square-root radial-transfer factor while retaining the same transported directional scale. The local estimates extend to compatible spherical triangulations. As an application, they yield an anisotropic energy-error estimate for the conforming approximation of the mean-zero Laplace--Beltrami problem with exact integration.

math.NA↗
arXiv2026

High-frequency coercivity loss for completely monotone memory on bounded time intervals

We study Volterra memory terms with locally integrable completely monotone kernels on a finite time interval $(0,\Tend)$ and ask how much $L^{2}$ coercivity they retain at high temporal frequencies. The main tool is an exact formula for the Rayleigh quotients of the cosine modes $ψ_n(t)=(2/\Tend)^{1/2}\cos(2πnt/\Tend)$. It shows that these quotients lie between $(1-κ_n/(2πn))\,m(2πn/\Tend)$ and $m(2πn/\Tend)$, where $m$ is the real memory symbol and $κ_n\in[1-e^{-2πn},1]$ is the best constant valid for all such kernels. Consequently, for non-constant kernels the algebraic decay rate of the quotients does not depend on $\Tend$ and coincides with the decay index $ρ\in[0,2]$ of $m$; any value in $[0,2]$ occurs. The Gaussian kernel $e^{-t^{2}}$, which is of positive type but not completely monotone, shows that this may fail otherwise: its quotients decay like $n^{-4}$, while its symbol decays faster than any power. We also show that the largest Rayleigh quotient on $(0,\Tend)$ is a continuous and strictly increasing function of $\Tend$, so that a kernel of total mass larger than one has exactly one critical horizon. Finally, because the memory operator on $(0,\Tend)$ is compact, it does not contribute to the uniform $L^{2}$ coercivity constant, and for $k_n(t)=ne^{-nt}$ the associated operators converge to the identity strongly but not in norm. The diffusion equation with memory serves as a model throughout.

math.AP↗
arXiv2026

Crouzeix--Raviart--Marini realisation of computable a priori $L^2$-error bounds on anisotropic meshes

We consider conforming $\Pone$ and lowest-order Crouzeix--Raviart (CR) approximations of the Dirichlet Poisson problem in two and three dimensions. We obtain computable a priori $L^2$-error bounds on anisotropic simplicial meshes without a global $H^2$-regularity assumption. For piecewise constant data, the Marini relation gives an equilibrated lowest-order Raviart--Thomas (RT) flux from the scalar CR solution. The square of the resulting Marini constant is the largest generalised eigenvalue of a problem involving only scalar conforming and CR matrices. Combined with the elementwise Poincaré constant, it controls both the conforming and CR errors. The CR $L^2$ estimate uses positivity of the discrete CR--conforming gap operator rather than a nonconforming Aubin--Nitsche argument. We also derive an exact decomposition of the Marini defect into the CR--conforming energy gap and an explicit geometric term with no aspect-ratio factor. On the algebraically graded L-shaped mesh family considered here, $q>3/2$ yields $κ_{M,h}=O(h)$ and computable $O(h^2)$ $L^2$-operator bounds.

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arXiv2026

Stable Reduction of Unresolved Dynamics: An Operator-Theoretic Framework

Dimension reduction removes variables, but a reliable reduction should not erase their dynamical influence. We study this principle for linear block evolution systems on resolved and unresolved Hilbert spaces. Exact elimination of the unresolved component yields a Volterra equation with a memory kernel and an effective forcing term carrying hidden initial data and forcing; in the Laplace domain the same operation is a dynamic Schur complement. We introduce finite-horizon trajectory-stable reduction and derive perturbation estimates for operator-valued kernels, then lift them to perturbations of the hidden propagator, couplings, hidden initial state, and hidden forcing. For passive skew-adjoint couplings, the reduced memory operator has positive type and an exact storage identity, and norm-convergent passive realisations preserve both trajectories and the storage--dissipation functional. Compatible finite spectral truncations provide structure-preserving internal-variable reductions. Two elementary examples separate positive type from instantaneous $L^2$-coercivity and show that small hidden amplitude need not imply small long-time influence. The framework isolates the controlled approximation of unresolved influence, rather than elimination alone, as the central reduction problem.

math.OC↗
arXiv2026

Exact characterisation of maximum-angle conditions for spherical finite element meshes

Maximum-angle conditions are standard finite-element mesh hypotheses that permit anisotropic triangles excluded by minimum-angle or shape-regularity assumptions. For exact spherical triangles, however, the angles of the chordal affine core and the intrinsic spherical angles need not coincide, while radial geometry introduces curvature-scale distortion. We give an algebraic characterisation of the intrinsic spherical maximum-angle condition through a dimensionless quantity computed from the three vertex vectors. A uniform positive lower bound on this quantity is equivalent to a uniform spherical maximum-angle bound and requires neither spherical-angle nor spherical-area evaluation. We identify the support-plane geometry linking the chordal circumradius to radial distortion and derive sharp comparisons between the intrinsic spherical and chordal semi-regularity parameters. In particular, a locality-independent comparison holds with sharp constant $2/\sqrt3$ and a characterised equality case. An area-based parameter is shown to be smaller than the spherical semi-regularity parameter, with sharp constant one in the local flat limit. For spherical finite-element meshes, the vertex criterion implies uniform chordal semi-regularity with an explicit constant, while relative refinement makes the radial-distortion factors converge uniformly to one. Thus, the intrinsic criterion, together with relative refinement, provides the geometric controls used in anisotropic finite-element analysis without imposing a minimum-angle or shape-regularity condition.

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arXiv2026

Weighted well-posedness and kernel stability for coercive evolution equations with measure-valued delays

We consider coercive evolution equations with measure-valued delay on a Gelfand triple. The delayed feedback is induced by a bounded form on $V$ and may contain principal spatial derivatives, so it is naturally $V^*$-valued rather than bounded on the pivot space. For every finite signed Borel kernel with no atom at the origin, an exponential weight makes the causal history operator contractive relative to the coercive parabolic solution operator. This yields finite-time well-posedness without a smallness condition on the total variation and without any positivity assumption on the delayed form. An asymmetric residual identity gives a total-variation Lipschitz estimate for signed kernels and strong stability for non-negative retarded kernels under narrow, equivalently weak, convergence of finite measures. A delayed-diffusion realisation shows that the framework genuinely covers principal-part delays, and narrow stability yields qualitative distributed-to-discrete convergence at every fixed positive lag.

math.AP↗
arXiv2026

Graph-space well-posedness for diffusion equations with degenerate instantaneous diffusion

We study diffusion equations with completely monotone memory when the instantaneous diffusion form is merely non-negative and may therefore lose coercivity. For a kernel whose Bernstein representing measure has finite total mass $M_{0}=ν([0,\infty))$, we introduce an extended state consisting of the physical variable and its continuum of internal variables. The aggregation and constant-embedding operators are adjoint with respect to the memory energy, and the resulting cross-term cancellation makes the augmented generator $m$-dissipative. This yields a unique mild solution, Lipschitz dependence on the data, and a contraction estimate that contains no positive lower bound for the instantaneous form. The zero-prehistory trajectories form a memory graph space, in which the problem is well posed in the sense of Hadamard. If, in addition, the first Bernstein moment $M_{1}=\int_{[0,\infty)}λ\,\diffν(λ)$ is finite, the memory potential and first-moment field possess the regularity needed to identify the semigroup solution with an encoded weak formulation and to obtain explicit stability bounds. We further prove uniform norm-resolvent convergence and convergence of the associated semigroups when a coercive instantaneous contribution vanishes. Under an additional $L^{2}(0,\Tend;V)$-regularity assumption on the limiting solution, the convergence rate in the memory graph norm is $O(\varepsilon^{1/2})$. These results provide a continuous stability target for structure-preserving and certified discretisations of memory-dominated diffusion.

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arXiv2026

Low-order CR--RT equilibrated-flux certification for semilinear problems on anisotropic meshes

We develop a low-order Crouzeix--Raviart--Raviart--Thomas (CR--RT) equilibrated-flux certification workflow for finite element approximations of semilinear diffusion--reaction problems, with particular emphasis on anisotropic mesh settings. Given a computed conforming finite element state $\tilde u_h$, the certification process is reduced to three computable quantities required by a Newton--Kantorovich argument: a dual-norm residual bound, a stability constant for the Fréchet derivative, and a Lipschitz bound for the derivative in a neighborhood of $\tilde u_h$. These components yield an explicit radius $ρ>0$, ensuring that the exact solution exists locally and uniquely within the ball $B(\tilde u_h,ρ)\subset V$. The residual bound is obtained from an $H(\mathrm{div})$-conforming $\mathbb{RT}^0$ certificate flux reconstructed through a Marini-type CR--RT route. The purpose of this route is not to replace general higher-order or local mixed equilibrated reconstructions, but to provide an explicit low-order construction whose algebraic structure is transparent on anisotropic simplicial meshes. Within the certified neighborhood, we further enclose selected quantities of interest $\mathcal J(u)$; the baseline enclosure follows from the verified inclusion, while an adjoint-based correction sharpens the resulting intervals. The numerical experiments report the behavior of the computable certification quantities for monotone semilinear models, including anisotropic mesh tests. Unless interval or outward-rounded scalar post-processing is explicitly used, the reported computations should be understood as floating-point evaluations of the derived rigorous estimators.

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arXiv2026

Exact-curved Lagrange finite elements for the Poisson problem in two dimensions

We develop an exact-curved Lagrange finite element framework for the Poisson problem on two-dimensional curved domains. The element map is factorised as $ F_K=Ψ_K\circΦ_{T_K}$, where $Φ_{T_K}$ maps the reference triangle to an affine core and $Ψ_K$ maps the affine core to the physical curved element. This factorisation separates affine scaling from curvature effects and allows the interpolation analysis to be carried out first on the affine core and then transferred to the exact curved element. For conforming linear Lagrange elements, we prove local $L^2$- and $H^1$-interpolation estimates on exact curved triangles. The estimates are expressed in terms of transported directional derivatives on the physical element, and the constants are independent of the anisotropic shape of the affine core under the stated semi-regularity assumptions. These interpolation estimates are then applied to derive energy-norm and $L^2$-error estimates for the Poisson problem. Numerical results on the unit disk illustrate the difference between straight-sided and curved geometric representations: the curved geometry reduces the geometric error substantially, while the leading finite element error remains governed by the $\mathbb{P}^1$ approximation.

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arXiv2026

On discrete Sobolev inequalities for nonconforming finite elements under a semi-regular mesh condition

We derive a discrete $ L^q-L^p$ Sobolev inequality tailored for the Crouzeix--Raviart and discontinuous Crouzeix--Raviart finite element spaces on anisotropic meshes in both two and three dimensions. Subject to a semi-regular mesh condition, this discrete Sobolev inequality is applicable to all pairs $(q,p)$ that align with the local Sobolev embedding, including scenarios where $q \leq p$. Importantly, the constant is influenced solely by the domain and the semi-regular parameter, ensuring robustness against variations in aspect ratios and interior angles of the mesh. The proof employs an anisotropy-sensitive trace inequality that leverages the element height, a two-step affine/Piola mapping approach, the stability of the Raviart--Thomas interpolation, and a discrete integration-by-parts identity augmented with weighted jump/trace terms on faces. This Sobolev inequality serves as a mesh-robust foundation for the stability and error analysis of nonconforming and discontinuous Galerkin methods on highly anisotropic meshes.

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arXiv2025

Nitsche's method under a semi-regular mesh condition

Nitsche's method is a numerical approach that weakly enforces boundary conditions for partial differential equations. In recent years, Nitsche's method has experienced a revival owing to its natural application in modern computational methods, such as the cut and immersed finite element methods. This study investigates Nitsche's methods based on an anisotropic weakly over-penalized symmetric interior penalty method for Poisson and Stokes equations on convex domains. As our primary contribution, we provide a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relationship between the Crouzeix and Raviart finite element space and the Raviart--Thomas finite element space. We present the error estimates in the energy norm on anisotropic meshes. We compared the calculation results for the anisotropic mesh partitions in the numerical experiments.

math.NA↗
arXiv2025

Anisotropic modified Crouzeix-Raviart finite element method for the stationary Navier-Stokes equation

We studied an anisotropic modified Crouzeix--Raviart finite element method for the rotational form of a stationary incompressible Navier--Stokes equation with large irrotational body forces. We present an anisotropic $H^1$ error estimate for the velocity of the modified Crouzeix--Raviart finite element method for the Navier--Stokes equation. The modified Crouzeix--Raviart finite element scheme was obtained using a lifting operator that mapped the velocity test functions to $H(÷;Ω)$-conforming finite element spaces. Because no shape-regularity mesh conditions are imposed, anisotropic meshes can be used for the analysis. The core idea of the proof involves using the relation between the Raviart--Thomas and Crouzeix--Raviart finite element spaces. Furthermore, we present a discrete Sobolev inequality under semi-regular mesh conditions to estimate the stability of the proposed method, and confirm the results obtained through numerical experiments.

math.NA↗
arXiv2024

Hybrid weakly over-penalised symmetric interior penalty method on anisotropic meshes

In this study, we investigate a hybrid-type anisotropic weakly over-penalised symmetric interior penalty method for the Poisson equation on convex domains. Compared with the well-known hybrid discontinuous Galerkin methods, our approach is simple and easy to implement. Our primary contributions are the proposal of a new scheme and the demonstration of a proof for the consistency term, which allows us to estimate the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart--Thomas finite element space and a discontinuous space. In numerical experiments, we compare the calculation results for standard and anisotropic mesh partitions.

math.NA↗
arXiv2024

Morley finite element analysis for fourth-order elliptic equations under a semi-regular mesh condition

In this study, we present a precise anisotropic interpolation error estimate for the Morley finite element method (FEM) and apply it to fourth-order elliptic equations. We do not impose the shape-regularity mesh condition in the analysis. Anisotropic meshes can be used for this purpose. The main contributions of this study include providing a new proof of the term consistency. This enables us to obtain an anisotropic consistency error estimate. The core idea of the proof involves using the relationship between the Raviart--Thomas and Morley finite-element spaces. Our results indicate optimal convergence rates and imply that the modified Morley FEM may be effective for errors.

math.NA↗
arXiv2024

Anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation

In this study, we investigate an anisotropic weakly over-penalised symmetric interior penalty method for the Stokes equation {on convex domains}. Our approach is a simple discontinuous Galerkin method similar to the Crouzeix--Raviart finite element method. As our primary contribution, we show a new proof for the consistency term, which allows us to obtain an estimate of the anisotropic consistency error. The key idea of the proof is to apply the relation between the Raviart--Thomas finite element space and a discontinuous space. While inf-sup stable schemes of the discontinuous Galerkin method on shape-regular mesh partitions have been widely discussed, our results show that the Stokes element satisfies the inf-sup condition on anisotropic meshes. Furthermore, we provide an error estimate in an energy norm on anisotropic meshes. In numerical experiments, we compare calculation results for standard and anisotropic mesh partitions.

math.NA↗
arXiv2022

Anisotropic Raviart-Thomas interpolation error estimates using a new geometric parameter

We present precise Raviart-Thomas interpolation error estimates on anisotropic meshes. The novel aspect of our theory is the introduction of a new geometric parameter of simplices. It is possible to obtain new anisotropic Raviart-Thoma error estimates using the parameter. We also include corrections to an error in "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 3 was incorrect.

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