In this paper, we propose and analyze a stabilized lower-order and equal-order virtual element method for the Stokes eigenvalue problem. The proposed method employs the same lowest-order conforming virtual element space and the same degrees of freedoms for both velocity and pressure. By adopting a mesh-dependent stabilization term associated with the pressure to the local bilinear form, we show the inf-sup condition and derive the stability for the source problem. We moreover employ the discrete spectral theory of compact operators to study the spectral approximation and obtain the optimal order for the eigenvalues. The proposed stabilized conforming lowest-order VEM of the Stokes eigenvalue problem is new, to the authors’ knowledge. Numerical results on polygonal and polyhedral meshes are presented to illustrate the performance of the proposed method and the theoretical results obtained.
In this paper, a trigonometric interpolation-based method is used to solve general non-linear ODE systems with certain constraints due to conservation laws in physics, biology, and engineering. In the case that a target solution of ODE is highly oscillated or rapid changed somewhere in its domain, the algorithm is expected to outperform significantly difference-based methods such as Runge-Kutta rules, which has been confirmed by several numerical experiments conducted in this paper.
In this paper, we present a deep Nitsche method for elliptic interface problems involving implicit jump conditions. To approximate the solution with discontinuities across the interface, we employ neural networks augmented with an additional variable to label the subdomains divided by the interface. The neural network solution is obtained by minimizing an energy functional derived from Nitsche’s formula. This approach handles boundary and interface conditions more effectively than typical deep Ritz methods. We establish an optimal error estimate of the proposed method under an inverse inequality assumption on the neural network functions, in the sense that the piecewise $H^1$-error can be bounded above by the best approximation error for the neural networks of the same class. Several numerical experiments are provided to demonstrate the performance of the proposed method.
This paper proposes two semi-implicit numerical schemes, LS-I and LS-II, for generalized Landau-Lifshitz (LL) equation with Dzyaloshinskii-Moriya interactions (DMI) and spin-transfer torque (STT) to simulate magnetic skyrmion dynamics. The schemes combine the scalar auxiliary variable (SAV) method with the midpoint rule to achieve first- and second-order temporal accuracy, respectively. Crucially, they are designed as structure-preserving methods that inherently maintain two fundamental physical properties of the fundamental LL equation: the constant magnitude of magnetization and the energy dissipation law. We rigorously prove that these schemes unconditionally preserve both properties under realistic boundary conditions for interfacial and bulk DMI respectively. A key implementation advantage stems from the SAV formulation, which decouples the non-local stray field, requiring only a single evaluation per time step and yielding a computational speedup of 3 to 5 times compared to conventional structure-preserving methods. Numerical experiments demonstrate the effectiveness of the proposed schemes in simulating the dynamics of magnetic skyrmions driven by current and external field.
This paper proposes a gradient-based neural network (GNN) framework for solving coupled matrix equations subject to bisymmetric constraints. Unlike existing neural solvers that do not explicitly preserve structural invariance, the proposed model embeds the bisymmetric operators directly into both the residual formulation and the gradient dynamics, ensuring that all trajectories evolve within the admissible bisymmetric subspace. A Lyapunov-based analysis establishes global convergence, and when equipped with nonlinear activation functions, the proposed GNN achieves fixed-time convergence independent of initial conditions. Numerical experiments on static, time-varying, and large-scale problems demonstrate improved convergence behavior compared with linear GNN counterparts. Applications to dynamical system analysis and constrained color image restoration demonstrate the potential of the proposed framework for handling structured matrix equations.
