In this paper, we propose an efficient spectral-Galerkin method for the quad-curl problem in a spherical domain. By introducing an auxiliary variable, we develop a mixed formulation that rigorously enforces the divergence-free constraint on the vector field. Leveraging the orthogonality of vector spherical harmonics and the technique of separation of variables, this mixed formulation is subsequently decoupled into a set of mutually independent one-dimensional fourth-order problems, corresponding to transverse-electric (TE) and transverse-magnetic (TM) modes, respectively. For the TE mode, we construct appropriate weighted Sobolev spaces and their finite-dimensional approximations, establish the associated variational formulation and discrete scheme, and present a rigorous error analysis based on the approximation properties of projection operators. For the TM mode, we introduce essential polar conditions to address the singularity of the problem and develop the corresponding variational formulation and discrete scheme. Numerical experiments are presented to demonstrate the effectiveness and spectral accuracy of the proposed method.
This study develops a residual-to-error analysis for second-order Singularly Perturbed Volterra Integro-Differential Equations (SPVIDEs) with boundary conditions. Building on the established auxiliary-output PINN principle, the Volterra integral is represented by an auxiliary network output, and the primary and auxiliary approximations are constrained through two coupled residuals. The novelty of the present work does not lie in this auxiliary-output construction itself, but in the analytical chain linking the auxiliary residual to Volterra-defect control and, subsequently, to a conditional ℒ^∞ solution-error estimate. More precisely, the Volterra defect is defined as the discrepancy between the learned auxiliary output and the Volterra operator applied to the learned primary solution. We show that this defect is directly controlled by the auxiliary residual. Combining the resulting defect estimate with the primary-equation residual and a generalized Grönwall–Bellman argument yields a problem-specific conditional ℒ^∞ bound whose constants explicitly reflect the effects of the perturbation parameter, the coupling coefficient, and the kernel bound. For numerical assessment, the boundary conditions and the auxiliary initial condition are imposed exactly through hard-constrained trial functions. A dual-output PINN is trained using Adam followed by L-BFGS, and its performance is evaluated through discrete maximum errors and worst-case errors over the tested perturbation set. The results show high accuracy for smooth and moderately perturbed problems. In contrast, the present global PINN is not robust in the strongest boundary-layer regimes, where the maximum errors remain comparatively large and do not decrease monotonically under collocation refinement.
We study the time-dependent Navier–Stokes system coupled with the heat equation under nonlinear Tresca boundary conditions in two dimension. This frictional slip boundary condition is characterized by a subdifferential property. Employing an effective regularization technique, we reformulate the continuous variational inequality into an equivalent equality problem. We establish the existence and and conditional uniqueness of weak solutions. Furthermore, we develop a fully discrete scheme based on the characteristic method and provide optimal error estimates for velocity, pressure, and temperature. We end the paper with numerical simulation validating the theoretical results.
We present a monolithic divergence-conforming virtual element method for the linear fluid–structure interaction (FSI) problem with a thick structure. This problem consists of the time-dependent Stokes equations in the fluid domain coupled with the linear elastodynamics equations in the solid region. To keep the divergence be conforming, the global velocity is discretized by the H(div) virtual element with an additional polynomial space defined on the element edges to approximate the tangential trace of the velocity, while the pressure is discretized by the discontinuous piecewise polynomials. Such spatial approximation combined with the Crank-Nicolson technique for the temporal discretization leads to a fully discrete scheme that produces an exactly divergence-free fluid velocity approximation. The a priori error estimates of the fully discrete scheme are well established, where the error bound is robust to the physical parameters. Finally, numerical examples are provided to verify that our method is not affected by the added mass effect and is robust to physical parameters. MSC codes are 65N30, 65N12.
