数值分析
In this paper, efficient alternating direction implicit (ADI) schemes are proposed to solve three-dimensional heat equations with irregular boundaries and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a modified ADI…
We are concerned in designing a suitable numerical scheme based on the equal-order hybrid high-order (HHO) method for the linear parabolic integro-differential equations. The spatial discretization is made using the equal-order HHO method…
We study the finite element approximation of linear second-order elliptic partial differential equations in nondivergence form with highly heterogeneous diffusion and drift coefficients. A generalized Cordes condition is imposed to…
We introduce a Hybrid High-Order (HHO) method for the Schr\"odinger equation in the presence of a magnetic vector potential. In quantum mechanics, physical observables are invariant under continuous gauge transformations, which must be kept…
Scattering resonances arise in wave phenomena and play an important role in many applications. While extensive theoretical studies have been conducted, effective numerical computation remains limited, and most existing methods suffer from…
The paper considers the problem of finding a submatrix $X_{\mathcal{S}} \in \mathbb{R}^{m \times k}$ in a matrix $X \in \mathbb{R}^{m \times n}$, such that the spectral or Frobenius norm of $X_{\mathcal{S}}^{\dag} X$ is limited, which…
Range functions are a fundamental tool for certified computations in geometric modeling, computer graphics, and robotics, but traditional range functions have only quadratic convergence order ($m=2$). For ``superior'' convergence order…
In 1985, Grassmann, Taksar, and Heyman published their celebrated paper, in which they introduced a numerically stable algorithm for computing the stationary probabilities of a finite-state Markov chain, one of the key performance…
The paper is concerned with three types of cubic splines over a triangulation that are characterized by three degrees of freedom associated with each vertex of the triangulation. The splines differ in computational complexity, polynomial…
The following hypothesis was formulated by Goreinov, Tyrtyshnikov, and Zamarashkin in \cite{goreinov1997theory}. If $U$ is $n\times k$ real matrix with the orthonormal columns $(n>k)$, then there exists a submatrix $Q$ of $U$ of size…
Neural network methods for PDEs require reliable error control in function space norms. However, trained neural networks can typically only be probed at a finite number of point values. Without strong assumptions, point evaluations alone do…
Studying the optoelectronic structure of materials can require the computation of several thousands of the smallest positive eigenpairs of a pseudo-hermitian Hamiltonian. Iterative eigensolvers may be preferred over direct methods for this…
We propose a structure-preserving parametric approximation for geometric flows with general anisotropic effects. By introducing a hyperparameter $\alpha$, we construct a unified surface energy matrix $\hat{\boldsymbol{G}}_k^\alpha(\theta)$…
This paper develops divergence-free mixed finite element methods for the Stokes equation. Using H(div)-conforming velocities and discontinuous pressures ensures the inf-sup condition for the velocity--pressure pair and yields pointwise…
Tensor product function (TPF) approximations have been widely adopted in solving high-dimensional problems, such as partial differential equations and eigenvalue problems, achieving desirable accuracy with computational overhead that scales…
The Lindblad equation is a widely used quantum master equation to model the dynamical evolution of open quantum systems whose states are described by density matrices. These solution matrices are characterized by semi-positiveness and trace…
Simulating infiltration in porous media using Richards' equation remains computationally challenging due to its parabolic structure and nonlinear coefficients. While a wide range of numerical methods for differential equations have been…
We consider the simplicial de Rham complex and the \v{C}ech-de Rham complex, two bigraded Hilbert complexes whose Hodge-Laplace problems govern spatially coupled problems in mixed dimension and homogeneous dimension, respectively. The…
We present a higher-order extension of the dual cell method for the time-domain Maxwell equations in three spatial dimensions. The approach builds upon a variational reinterpretation of the Finite Integration Technique on dual meshes and…
In this paper, we propose an efficient two-level additive Schwarz method for solving large-scale eigenvalue problems arising from the finite element discretization of symmetric elliptic operators, which may compute efficiently more interior…