arXiv subjects

Tianyang Chu

Publications and source records attributed to Tianyang Chu.

5 recordsLinked to original sources

arXiv2026

Numerical analysis of linear finite element methods for the Gross--Pitaevskii eigenvalue problem

In this paper, we analyze the standard and the mass-lumped linear finite element methods for the Gross--Pitaevskii eigenvalue problem. For the standard method, we give a new proof of the uniqueness of the discrete ground state up to sign for sufficiently small mesh sizes, which only requires the $H^1$-convergence of the discrete ground states. For both methods, we establish optimal convergence rates in the $L^p$- and $W^{1,p}$-norms for $1 \le p < \infty$. Under additional regularity assumptions on the potential, we further obtain an optimal convergence rate in the $W^{1,\infty}$-norm and an almost optimal convergence rate, up to a logarithmic factor, in the $L^\infty$-norm. Moreover, we prove that the discrete ground states superconverge to the Ritz projection of the exact ground state in the $W^{1,p}$-seminorm. The analysis is based on a perturbation argument. Numerical experiments confirm the theoretical results.

math.NA↗

arXiv2026

A recovery based linear finite element method for the Monge--Ampère equation

In this paper, we study finite element approximations of a fourth-order perturbed semilinear partial differential equation that arises naturally in the computation of viscosity solutions to the fully nonlinear Monge--Ampère equation. We develop a novel recovery-based $C^0$ linear finite element method for discretizing this model problem. The key analytical tool is a Miranda--Talenti-type estimate for the recovery operator, which guides the formulation of the discrete scheme and guarantees its well-posedness. We rigorously establish convergence in the $H^2$ norm by employing a fixed-point argument. Furthermore, optimal $H^1$-error estimates are derived using a duality argument. Numerical experiments confirm the efficiency of the proposed method and validate the theoretical convergence results.

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arXiv2025

An orthogonality-preserving approach for eigenvalue problems

Solving large-scale eigenvalue problems poses a significant challenge due to the computational complexity and limitations on the parallel scalability of the orthogonalization operation, when many eigenpairs are required. In this paper, we propose an intrinsic orthogonality-preserving model, formulated as an evolution equation, and a corresponding numerical method for eigenvalue problems. The proposed approach automatically preserves orthogonality and exhibits energy dissipation during both time evolution and numerical iterations, provided that the initial data are orthogonal, thus offering an accurate and efficient approximation for the large-scale eigenvalue problems with orthogonality constraints. Furthermore, we rigorously prove the convergence of the scheme without the time step size restrictions imposed by the CFL conditions. Numerical experiments not only corroborate the validity of our theoretical analyses but also demonstrate the remarkably high efficiency of the algorithm.

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arXiv2025

A gradient flow model for the Gross--Pitaevskii problem: Mathematical and numerical analysis

This paper concerns the mathematical and numerical analysis of the $L^2$ normalized gradient flow model for the Gross--Pitaevskii eigenvalue problem, which has been widely used to design the numerical schemes for the computation of the ground state of the Bose--Einstein condensate. We first provide the mathematical analysis for the model, including the well-posedness and the asymptotic behavior of the solution. Then we propose a normalized implicit-explicit fully discrete numerical scheme for the gradient flow model, and give some numerical analysis for the scheme, including the well-posedness and optimal convergence of the approximation. Some numerical experiments are provided to validate the theory.

math.NA↗

arXiv2024

New $C^0$ interior penalty method for Monge-Ampère equations

Monge-Ampère equation is a prototype second-order fully nonlinear partial differential equation. In this paper, we propose a new idea to design and analyze the $C^0$ interior penalty method to approximation the viscosity solution of the Monge-Ampère equation. The new methods is inspired from the discrete Miranda-Talenti estimate. Based on the vanishing moment representation, we approximate the Monge-Ampère equation by the fourth order semi-linear equation with some additional boundary conditions. We will use the discrete Miranda-Talenti estimates to ensure the well-posedness of the numerical scheme and derive the error estimates.

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