ABSTRACT A unified high‐order spectral–pseudospectral framework was proposed for the efficient numerical solution of nonlinear fractional partial differential equations (FPDEs) on irregular and curved domains. The method combined eigenfunction‐based spectral discretizations with grid‐based pseudospectral operators to accurately resolve smooth solutions, strong nonlinearities, and nonlocal fractional operators. Complex geometries were treated using domain mappings, spectral elements, and a flexible fictitious‐domain embedding with penalty or masking strategies that preserved spectral convergence. Stable time integration was achieved through exponential time‐differencing and energy‐stable implicit–explicit schemes tailored to fractional diffusion, dispersive, and reaction–diffusion dynamics. The framework was applied to fractional diffusion equations, fractional Schrödinger models, Fisher‐type systems, and the Schnakenberg activator–inhibitor model, where long‐range interactions and fine‐scale pattern formation were successfully captured. Rigorous stability and convergence estimates were established in fractional Sobolev norms. Numerical experiments on benchmark problems with complex geometries confirmed spectral accuracy, excellent agreement with reference solutions, and high computational efficiency. The approach was shown to naturally extend to three‐dimensional and manifold domains through FFT‐accelerated, matrix‐free implementations.
ABSTRACT In this article, the numerical analysis of the Jacobi spectral collocation method for high‐dimensional volume‐constrained nonlocal models with weakly singular kernels is established. A rigorous convergence analysis has been established to show that the approximate solution of the method can achieve spectral accuracy for sufficiently smooth solutions in the sense of norm. Moreover, the asymptotically compatible property of the numerical method has been proved, that is, we show that the numerical solution can converge to its correct local limit as nonlocal interactions vanish. Finally, some numerical examples are shown to support the convergence analysis.
ABSTRACT In this article, we present a numerical analysis of the extended Fisher–Kolmogorov equation and use a mixed finite element method for its approximation. Our analysis begins with establishing the existence and uniqueness of the weak solution based on the Faedo–Galerkin approach. Next, for the fully discrete scheme, a Crank–Nicolson time discretization is employed. The stability estimate is then derived using an appropriate Lyapunov functional, and subsequently the existence of the fully discrete solution is proved using Brouwer's fixed‐point theorem. The error analysis demonstrates second‐order convergence for both the solution variable and . Our study also investigates the influence of the fourth‐order term in the extended Fisher–Kolmogorov equation, highlighting its stabilizing effects and its role in the dynamics of front propagation. Finally, numerical experiments are presented to confirm the theoretical results and to illustrate the effectiveness and reliability of the proposed scheme.
ABSTRACT In this paper, we propose a shallow neural network method specifically designed for solving partial differential equations (PDEs) on surfaces, called the Geometry‐Aware Transferable Neural Network (GATransNet). The method effectively addresses the challenge of determining hidden‐layer neurons in fixed‐feature networks. By introducing a geometry‐aware selection strategy, our approach closely integrates the geometric characteristics of neurons with the tangent‐plane structure of the surface, significantly enhancing the model's adaptability and stability. Furthermore, we combine the backward differentiation formula in time with the proposed GATransNet in space to construct a fully discrete numerical scheme, termed BDF‐GATransNet, for solving time‐dependent problems. Extensive numerical experiments involving the Laplace–Beltrami operator and related surface PDEs on both static and evolving surfaces are conducted to demonstrate the accuracy, efficiency, and flexibility of the proposed method. These results highlight its potential as a robust and versatile approach for solving surface PDEs.
ABSTRACT We propose, analyze, and investigate numerically a novel two‐level Galerkin reduced order model (2L‐ROM) for the efficient and accurate numerical simulation of nonlinear problems. In the first step, a relatively low‐dimensional nonlinear system is solved. In the second step, the given nonlinear equations are linearized around the solution found in the first step, and the linearized problem is solved. Although, the new 2L‐ROM can be used for general nonlinear equations, in our theoretical investigation, we consider the steady Navier–Stokes equations. We prove that, for appropriate parameters, the 2L‐ROM error bound is of the same order as the error bound for the standard 1L‐ROM. Furthermore, we develop 2L‐ROM parameter scalings that balance the 2L‐ROM's numerical accuracy and computational efficiency. In our numerical investigation, we consider the Burgers equation, which is a simplified computational setting that is often used in the ROM assessment. Our numerical investigation shows that the new 2L‐ROM significantly decreases (by a factor of 2 and even close to 3) the 1L‐ROM computational cost, without compromising its numerical accuracy. Furthermore, we show that the 2L‐ROM parameters that yield the best numerical results are similar to those obtained from the theoretical scalings.
