Xiaobo Yin

Qiang Du, Xiaobo Yin

Numerical solution of nonlocal constrained value problems with integrable kernels are considered. These nonlocal problems arise in nonlocal mechanics and nonlocal diffusion. The structure of the true solution to the problem is analyzed first. The analysis leads naturally to a new kind of discontinuous Galerkin method that efficiently solve the problem numerically. This method is shown to be asymptotically compatible. Moreover, it has optimal convergence rate for one dimensional case under very weak assumptions, and almost optimal convergence rate for two dimensional case under mild assumptions.

Xiaobo Yin, Hehu Xie

A uniform strategy to derive metric tensors in two spatial dimension for interpolation errors and their gradients in $L^p$ norm is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in corresponding metric space, with the metric tensor being computed based on a posteriori error estimates in different norms. Numerical results show that the corresponding convergence rates are always optimal.

Xiaobo Yin, Hehu Xie

In this paper, we give a new type of a posteriori error estimators suitable for moving finite element methods under anisotropic meshes for general second-order elliptic problems. The computation of estimators is simple once corresponding Hessian matrix is recovered. Wonderful efficiency indices are shown in numerical experiments.

Yana Di, Hehu Xie, Xiaobo Yin

We propose a numerical strategy to generate the anisotropic meshes and select the appropriate stabilized parameters simultaneously for two dimensional convection-dominated convection-diffusion equations by stabilized continuous linear finite elements. Since the discretized error in a suitable norm can be bounded by the sum of interpolation error and its variants in different norms, we replace them by some terms which contain the Hessian matrix of the true solution, convective fields, and the geometric properties such as directed edges and the area of the triangle. Based on this observation, the shape, size and equidistribution requirements are used to derive the corresponding metric tensor and the stabilized parameters. It is easily found from our derivation that the optimal stabilized parameter is coupled with the optimal metric tensor on each element. Some numerical results are also provided to validate the stability and efficiency of the proposed numerical strategy.

Xiaobo Yin, Hehu Xie

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on a posteriori error estimates proposed in \cite{YinXie}. The new metric tensor explores more comprehensive information of anisotropy for the true solution than those existing ones. Numerical results show that this approach can be successfully applied to deal with poisson and steady convection-dominated problems. The superior accuracy and efficiency of the new metric tensor to others is illustrated on various numerical examples of complex two-dimensional simulations.

Xiuzhu Yang, Xiaobo Yin

The Improved Partial Area-Analytical Calculation (IPA-AC) method represents a leading meshfree discretization strategy for peridynamic models, distinguished by its rigorous geometric treatment of boundary intersections via dual corrections of integration weights and quadrature points. Despite its empirical success in suppressing boundary-induced geometric errors, a systematic theoretical characterization of its convergence behaviors under distinct scaling limits has remained elusive. This work establishes a unified convergence framework for the IPA-AC method applied to both scalar and tensor kernels. By leveraging the Lax Equivalence Theorem, we explicitly derive error estimates that reveal the method's performance across three critical limiting regimes. The theoretical analysis, substantiated by numerical validation, demonstrates that: (1) for a fixed horizon $δ$, the method achieves robust second-order convergence $\mathcal{O}(h ^{2})$ with respect to the mesh size $h$; (2) for a fixed mesh, the discretization error scales as $\mathcal{O}(δ^{-2})$, indicating a sensitivity to the nonlocal length scale; and (3) the method does not satisfy the Asymptotic Compatibility (AC) condition. These findings clarify that while the IPA-AC method offers superior accuracy for simulating fixed nonlocal models, it requires a sufficiently large horizon-to-mesh ratio to mitigate intrinsic discretization errors when approximating the local limit.

Qingkui Ma, Hehu Xie, Xiaobo Yin

Fractional PDEs involving the fractional Laplacian on bounded domains are challenging because of hypersingular nonlocal kernels, exterior Dirichlet constraints, reduced boundary regularity, and the high computational cost in high dimensions. To address these issues, we first adopt a spatially varying radius with directional distance-to-boundary information, which yields a geometry-adaptive three-part decomposition of the fractional Laplacian: singular near-field, regular interior far-field, and analytical exterior far-field contributions. Then we employ Gauss-Jacobi quadrature for the singular radial integral, Gauss quadrature for the regular interior radial integral, and Monte Carlo sampling for the angular variables. A feature-enhanced physics-informed neural network trial space is finally used to tackle the low-regularity behavior near the boundary. Through the above steps, we obtain a quadrature-enhanced Monte Carlo fractional physics-informed neural network (QE-MC-fPINN) method. Numerical experiments on fractional Poisson equations and time-dependent fractional PDEs show that, on the tested benchmarks, the proposed method outperforms two representative MC-fPINN discretizations in accuracy and convergence, especially for solutions with strong boundary singularities.

Zhongshuo Lin, Qingkui Ma, Hehu Xie et al.

In this paper, we propose a novel machine learning method based on adaptive tensor neural network subspace to solve linear time-fractional diffusion-wave equations and nonlinear time-fractional partial integro-differential equations. In this framework, the tensor neural network and Gauss-Jacobi quadrature are effectively combined to construct a universal numerical scheme for the temporal Caputo derivative with orders spanning $ (0,1)$ and $(1,2)$. Specifically, in order to effectively utilize Gauss-Jacobi quadrature to discretize Caputo derivatives, we design the tensor neural network function multiplied by the function $t^μ$ where the power $μ$ is selected according to the parameters of the equations at hand. Finally, some numerical examples are provided to validate the efficiency and accuracy of the proposed tensor neural network based machine learning method.