arXiv subjects

Tokuhiro Eto

Publications and source records attributed to Tokuhiro Eto.

10 recordsLinked to original sources

arXiv2026

A Structure-Preserving Method of Fundamental Solutions for the Multi-Phase Mullins-Sekerka Flow

A charge simulation method is applied to approximate the multi-phase Mullins--Sekerka flow in $\mathbb R^{2}$ and in a half-plane bounded by a Neumann wall. In the underlying mathematical model, interfaces driven by their curvature are coupled through a harmonic chemical-potential field. We use a charge simulation method, a variant of the method of fundamental solutions: each chemical potential is represented by fundamental solutions centered at charge points off the curve, so no bulk mesh is required. It treats curve networks separating several phases at triple junctions, including phases that occupy more than one region; on the half-plane boundary, the no-flux condition is imposed exactly by image charges, and mobile contacts stay orthogonal to the wall. The discretization is structure-preserving in the sense that every bounded phase area is conserved to machine precision at the velocity level by a null-space projection of the discrete area constraints. The proposed scheme is assessed through a convergence test against an exact three-concentric-circle solution.

math.NA↗

arXiv2026

Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain

This paper studies a Chambolle-type minimizing movement scheme for mean curvature flow with prescribed contact angle in a smooth bounded domain. The scheme is based on the capillary functional and the geodesic signed distance relative to the container, and yields a time-discrete level-set approximation. The main result asserts that, for every Lipschitz-continuous boundary function prescribing a strictly nondegenerate contact angle, the approximate solutions converge locally uniformly to the unique viscosity solution of the corresponding level-set mean curvature equation with oblique derivative boundary condition. This improves a previous convergence theorem, where the container was assumed to be convex and a curvature-type condition relating the tangential derivative of the prescribed contact-angle function to the principal curvatures of the container boundary was imposed. The main new ingredient is a uniform Lipschitz estimate for the solutions of the variational problems defining the scheme. This estimate is derived by applying a Bernstein-type argument to a suitable weighted gradient, rather than to the gradient itself, which rules out boundary maxima without relying on the previous curvature-type condition.

math.AP↗

arXiv2026

A Parametric Finite Element Approach for an Anisotropic Multi-Phase Mullins-Sekerka Problem with Kinetic Undercooling

We consider a sharp interface formulation for an anisotropic multi-phase Mullins-Sekerka problem with kinetic undercooling. The flow is characterized by a cluster of surfaces evolving such that the total surface energy plus a weighted sum of the volumes of the enclosed phases decreases in time. Upon deriving a suitable variational formulation, we introduce a fully discrete unfitted finite element method. In this approach, the approximations of the moving interfaces are independent of the triangulations used for the equations in the bulk. Our method can be shown to be unconditionally stable. Several numerical examples demonstrate the capabilities of the introduced method. In particular, it is demonstrated that the evolution of multiple ice crystals with junctions can be modeled using the proposed approach.

math.NA↗

arXiv2025

A hyperbolic finite difference scheme for anisotropic diffusion equations: preserving the discrete maximum principle

A hyperbolic system approach is proposed for robust computation of anisotropic diffusion equations that appear in quasineutral plasmas. Though the approach exhibits merits of high extensibility and accurate flux computation, the monotonicity of the scheme for anisotropic diffusion cases has not been understood. In this study, the discrete maximum principle (DMP) of the hyperbolic system approach is analyzed and tested in various anisotropic diffusion cases. A mathematical analysis is conducted to obtain an optimal condition of an arbitrary parameter to guarantee the DMP, and numerical experiments reveal an adoptive selection of the parameter for DMP-preserving results. It is confirmed that, with an appropriate preconditioning matrix and parameter choice, the hyperbolic system approach preserves the DMP even with a linear discretization.

math.NA↗

arXiv2025

A parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions

In this study, we propose a parametric finite element method for a degenerate multi-phase Stefan problem with triple junctions. This model describes the energy-driven motion of a surface cluster whose distributional solution was studied by Garcke and Sturzenhecker. We approximate the weak formulation of this sharp interface model by an unfitted finite element method that uses parametric elements for the representation of the moving interfaces. We establish existence and uniqueness of the discrete solution and prove unconditional stability of the proposed scheme. Moreover, a modification of the original scheme leads to a structure-preserving variant, in that it conserves the discrete analogue of a quantity that is preserved by the classical solution. Some numerical results demonstrate the applicability of our introduced schemes.

math.NA↗

arXiv2025

On existence and uniqueness for transport equations with non-smooth velocity fields under inhomogeneous Dirichlet data

A transport equation with a non-smooth velocity field is considered under inhomogeneous Dirichlet boundary conditions. The spatial gradient of the velocity field is assumed in $L^{p'}$ in space and the divergence of the velocity field is assumed to be bounded. By introducing a suitable notion of solutions, it is shown that there exists a unique renormalized weak solution for $L^p$ initial and boundary data for $1/p+1/p'=1$. Our theory is considered as a natural extension of the theory due to DiPerna and Lions (1989), where there is no boundary. Although a smooth domain is considered, it is allowed to be unbounded. A key step is a mollification of a solution. In our theory, mollification in the direction normal to the boundary is tailored to approximate the boundary data.

math.AP↗

arXiv2024

A structure-preserving finite element method for the multi-phase Mullins-Sekerka problem with triple junctions

We consider a sharp interface formulation for the multi-phase Mullins-Sekerka flow. The flow is characterized by a network of curves evolving such that the total surface energy of the curves is reduced, while the areas of the enclosed phases are conserved. Making use of a variational formulation, we introduce a fully discrete finite element method. Our discretization features a parametric approximation of the moving interfaces that is independent of the discretization used for the equations in the bulk. The scheme can be shown to be unconditionally stable and to satisfy an exact volume conservation property. Moreover, an inherent tangential velocity for the vertices on the discrete curves leads to asymptotically equidistributed vertices, meaning no remeshing is necessary in practice. Several numerical examples, including a convergence experiment for the three-phase Mullins-Sekerka flow, demonstrate the capabilities of the introduced method.

math.NA↗

arXiv2024

A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

We consider a minimizing movement scheme of Chambolle type for the mean curvature flow equation with prescribed contact angle condition in a smooth bounded domain in $\mathbb{R}^d$ ($d\geq2$). We prove that an approximate solution constructed by the proposed scheme converges to the level-set mean curvature flow with prescribed contact angle provided that the domain is convex and that the contact angle is away from zero under some control of derivatives of given prescribed angle. We actually prove that an auxiliary function corresponding to the scheme uniformly converges to a unique viscosity solution to the level-set equation with an oblique {derivative} boundary condition corresponding to the prescribed boundary condition.

math.AP↗

arXiv2023

On a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain and its computation

We introduce a capillary Chambolle type scheme for mean curvature flow with prescribed contact angle. Our scheme includes a capillary functional instead of just the total variation. We show that the scheme is well-defined and has consistency with the energy minimizing scheme of Almgren-Taylor-Wang type. Moreover, for a planar motion in a strip, we give several examples of numerical computation of this scheme based on the split Bregman method instead of a duality method.

math.NA↗

arXiv2023

A rapid numerical method for the Mullins-Sekerka flow with application to contact angle problems

The Mullins-Sekerka problem is numerically solved in $\mathbb{R}^2$ with the aid of the charge simulation method. This is an expansion of the numerical scheme by which Sakakibara and Yazaki computed the Hele-Shaw flow. We investigate a sufficient condition for the number of collocation points to ensure that the length of the generated approximate polygonal curves gradually decreases. We propose a new benchmark function for the Mullins-Sekerka flow to confirm that the scheme works well. Moreover, by changing the fundamental solutions of the charge simulation method, we are successful to establish a numerical scheme that can be used to treat the Mullins-Sekerka problem with the contact angle condition.

math.NA↗