This paper develops unconditional stability preserving reduced-dimension finite difference (RDFD) schemes for solving Caputo-type time-fractional diffusion equations (FDEs) of order α ∈ (0, 1) in one and two spatial dimensions. The Caputo derivative is discretized by the L2-1σ formula, and spatial derivatives are approximated using the central finite difference (FD), forming the L2-1σ-FD schemes as the foundational numerical frameworks. To reduce the computational complexity of high-dimensional FDEs, we further develop L2-1σ-RDFD schemes by integrating proper orthogonal decomposition (POD), achieving efficient dimension reduction while preserving numerical accuracy. We rigorously prove the unconditional stability and optimal convergence order of the proposed RDFD schemes. Numerical experiments confirm the theoretical results and demonstrate that the RDFD schemes significantly reduce computational cost without sacrificing accuracy compared to the original FD schemes. The proposed methods offer a general and efficient framework for numerically solving fractional diffusion models and are readily extensible to more complex fractional partial differential equations arising in interdisciplinary fields such as computational physics, engineering mechanics, and mathematical biology.
This paper is devoted to the analysis and numerical approximation of a class of nonlinear parabolic quasi-variational inequalities with solution-dependent nonlocal constraints. Unlike classical variational inequalities, the admissible set is determined by a nonlinear obstacle operator that depends explicitly on the evolving solution, resulting in a moving feasible region whose geometry changes throughout the evolution. This intrinsic coupling between the governing parabolic operator and the admissible set substantially increases both the analytical and computational complexity of the problem. We first investigate the fundamental analytical properties of the nonlocal obstacle operator and establish its monotonicity, continuity, Lipschitz stability, non-expansivity, and compactness under suitable structural assumptions. These results provide the analytical framework required for the study of the corresponding quasi-variational problem. By combining monotone operator techniques, compactness arguments, energy estimates, pseudomonotonicity methods, and nonlinear fixed-point theory, we establish the existence and uniqueness of weak solutions together with continuous dependence on the problem data. For the numerical approximation, we develop a fully discrete semi-implicit finite element method obtained by coupling a conforming spatial discretization with an implicit time-stepping procedure and an iterative obstacle projection algorithm that enforces the solution-dependent constraint at every temporal level. The fully discrete problem is shown to be well posed through discrete stability estimates, energy inequalities, and rigorous convergence analysis, leading to optimal a priori error estimates for the proposed approximation. Extensive two-dimensional numerical experiments are presented to validate the theoretical results. The computational study investigates spatial and temporal convergence, discrete energy evolution, nonlinear iteration behaviour, computational efficiency, algorithmic scalability, and robustness with respect to different nonlinear diffusion laws. The numerical results exhibit excellent agreement with the theoretical predictions, accurately preserve the solution-dependent obstacle constraint, and demonstrate that the proposed numerical framework provides an accurate, stable, and computationally efficient approach for solving nonlinear parabolic quasi-variational inequalities with nonlocal state-dependent constraints.
In this paper, a hybrid Generalized Finite Difference-Radial Basis Function (GFD-RBF) meshless method is developed for the incompressible Navier-Stokes equations. The method is applied to the three-sided lid-driven cavity flow in a square domain involving complex flow features, and results are obtained for Reynolds numbers up to 10, 000 for the first time. Additionally, the proposed approach captures a third secondary vortex at Re=1000 for the parallel motion of the opposite walls, representing a flow feature that has not been previously reported in the literature. The hybrid formulation preserves the complementary strengths of both methods, enabling a stable and accurate discretization of the governing equations. This is verified through the analysis of the convergence behavior of the resulting discretized system. The order of accuracy of the results obtained using the hybrid meshless method is also presented. Furthermore, the hybrid scheme exhibits favorable computational efficiency by achieving converged solutions on relatively coarse nodal distributions, highlighting its efficiency for nonlinear convection-dominated problems.
Euler’s classic observation (1757) on the equipollence between conservative body forces and the pressure stress reaction in incompressible media is critically examined and extended to the weak form of mixed boundary value problems. This paper demonstrates that when a conservative body force potential does not vanish on a Neumann boundary, a geometric modeling error occurs in standard formulations unless a corrective, potential-induced boundary traction is applied.To address this, a novel Auxiliary Principle of Virtual Work (AUX-PVW) is derived. While shown to be equivalent to the classical formulation at the continuous level under appropriate boundary regularisation, the AUX-PVW fundamentally differs in its mathematical structure. Specifically, it shifts the conservative body force from a gradient term to a pure scalar field, thereby relaxing the regularity requirements on the force potential. This regularization shift has profound implications for mixed Bubnov-Galerkin finite element methods, where the pressure reaction part of the standard tangent stiffness operator is merely shifted by the scalar potential.Finally, the theory is contextualized within Ericksen’s (1954) seminal class of controllable deformations. It is shown that the underlying kinematics and the self-similar evolution of curved material surfaces (such as S2 manifolds) remain invariant under controllable conservative fields, provided the companion non-symmetric load-stiffness contributions on the boundary are properly accounted for. Supporting analytical evidence is provided for the inflation of a thick spherical sector, while numerical validation of the classical formulation is relegated to a companion paper.
With the goal of accurately extracting the optical field losses in a three-dimensional (3D), circularly coiled waveguide (e.g., bent optical fiber), this effort presents the numerical methodologies that are implemented for an envelope Maxwell model that propagates electromagnetic fields as an entirely boundary value problem. Our unique modeling approach includes an ultraweak variational formulation of the envelope Maxwell model in the curved geometry of the bending, which is discretized by the discontinuous Petrov–Galerkin (DPG) method, which permits residual-driven mesh and polynomial-order adaptivity. This also, then, requires a unique approach for constructing perfectly matched layers (PMLs) as absorbing boundary conditions in both the direction of optical field propagation and in the tangential directions, where unguided energy escapes the waveguide. Our coiled waveguide modeling technology extracts the mode confinement losses from the propagation of the coherent optical field through the bent waveguide. We verify our simulations against the semi-analytical results from the analogous bent slab waveguide problem, and we successfully demonstrate stable convergence to loss values for the 3D coiled optical fiber problem, which has never been done previously for our specific modeling approach.
