Xianping Li

Weizhang Huang, Lennard Kamenski, Xianping Li

Anisotropic mesh adaptation has been successfully applied to the numerical solution of partial differential equations but little considered for variational problems. In this paper, we investigate the use of a global hierarchical basis error estimator for the development of an anisotropic metric tensor needed for the adaptive finite element solution of variational problems. The new metric tensor is completely a~posteriori and based on residual, edge jumps and the hierarchical basis error estimator. Numerical results show that it performs comparable with existing metric tensors based on Hessian recovery. A few sweeps of the symmetric Gauß-Seidel iteration for solving the global error problem prove sufficient to provide directional information necessary for successful mesh adaptation. .

Fei Zhang, Weizhang Huang, Xianping Li et al.

In the phase-field modeling of brittle fracture, anisotropic constitutive assumptions for the degradation of stored elastic energy due to fracture are crucial to preventing cracking in compression and obtaining physically sound numerical solutions. Three energy decomposition models, the spectral decomposition, the volumetric-deviatoric split, and an improved volumetric-deviatoric split, and their effects on the performance of the phase-field modeling are studied. Meanwhile, anisotropic degradation of stiffness may lead to a small energy remaining on crack surfaces, which violates crack boundary conditions and can cause unphysical crack openings and propagation. A simple yet effective treatment for this is proposed: define a critically damaged zone with a threshold parameter and then degrade both the active and passive energies in the zone. A dynamic mesh adaptation finite element method is employed for the numerical solution of the corresponding elasticity system. Four examples, including two benchmark ones, one with complex crack systems, and one based on an experimental setting, are considered. Numerical results show that the spectral decomposition and improved volumetric-deviatoric split models, together with the improvement treatment of crack boundary conditions, can lead to crack propagation results that are comparable with the existing computational and experimental results. It is also shown that the numerical results are not very sensitive to the parameter defining the critically damaged zone.

Xianping Li, Woinshet D. Mergia, Kailash C. Patidar

In ecological studies of pattern formation, models of the competitive-diffusion type are generally singularly perturbed, and the numerical approximation of such models is challenging. In this paper, we present finite element discretization combined with a second-order semi-implicit nonlinear multistep scheme for a two-dimensional three-species competition-diffusion system with distinct mobility rates. The method employs a $C^0$-conforming Galerkin finite element approximation in space and a Crank-Nicolson/Adams-Bashforth-type time integration that treats the diffusion terms implicitly while linearizing the nonlinear reaction terms in a stage-by-stage manner. The resulting scheme is linear at each time step and avoids iterative nonlinear solvers. Rigorous stability analysis shows that the discrete method inherits the asymptotic stability properties of the continuous model without restrictions on the time step size. Numerical simulations for various mobility regimes demonstrate the ability of the proposed method to capture complex spatio-temporal patterns, including droplet-like, banded, spiral, and glider-type structures.

Xianping Li, Matthew McCoy

Time-dependent convection-dominated convection-diffusion problems are considered. We develop a moving mesh streamline upwind Petrov-Galerkin (MM-SUPG) method by combining residual-based SUPG stabilization with a metric-based moving mesh PDE (MMPDE) approach. The key feature of the method is the interaction between mesh adaptation and stabilization: the evolving mesh modifies both the direction and magnitude of the SUPG stabilization through the local element geometry, while stabilization improves robustness in convection-dominated regimes. For isotropic diffusion, numerical results show that the proposed method reduces spurious oscillations and provides improved resolution of sharp layers compared with fixed mesh methods, while yielding accuracy comparable to moving mesh finite element methods without SUPG in some cases. For anisotropic diffusion, we introduce a weighted tensor that incorporates both the diffusion tensor and the convection field, and construct a metric tensor via intersection to guide mesh adaptation. Under suitable assumptions on the elementwise tensors and mesh geometry, we establish sufficient conditions for the discrete maximum principle (DMP) of the fully discrete scheme. The analysis is based on quantitative bounds for the convection terms, and requires structural conditions on the alignment between the velocity field and the diffusion tensor. Numerical experiments demonstrate that the proposed metric improves monotonicity properties and reduces undershoots, while maintaining overall accuracy.

Xianping Li, Teresa Wu

Compressed sensing (CS) has become a popular field in the last two decades to represent and reconstruct a sparse signal with much fewer samples than the signal itself. Although regular images are not sparse on their own, many can be sparsely represented in wavelet transform domain. Therefore, CS has also been widely applied to represent digital images. However, an alternative approach, adaptive sampling such as mesh-based image representation (MbIR), has not attracted as much attention. MbIR works directly on image pixels and represents the image with fewer points using a triangular mesh. In this paper, we perform a preliminary comparison between the CS and a recently developed MbIR method, AMA representation. The results demonstrate that, at the same sample density, AMA representation can provide better reconstruction quality than CS based on the tested algorithms. Further investigation with recent algorithms is needed to perform a thorough comparison.

Karrar Abbas, Xianping Li

As the resolution of digital images increase significantly, the processing of images becomes more challenging in terms of accuracy and efficiency. In this paper, we consider image segmentation by solving a partial differentiation equation (PDE) model based on the Mumford-Shah functional. We develop a new algorithm by combining anisotropic mesh adaptation for image representation and finite element method for solving the PDE model. Comparing to traditional algorithms solved by finite difference method, our algorithm provides faster and better results without the need to resizing the images to lower quality. We also extend the algorithm to segment images with multiple regions.

Xianping Li

Triangular meshes have gained much interest in image representation and have been widely used in image processing. This paper introduces a framework of anisotropic mesh adaptation (AMA) methods to image representation and proposes a GPRAMA method that is based on AMA and greedy-point removal (GPR) scheme. Different than many other methods that triangulate sample points to form the mesh, the AMA methods start directly with a triangular mesh and then adapt the mesh based on a user-defined metric tensor to represent the image. The AMA methods have clear mathematical framework and provides flexibility for both image representation and image reconstruction. A mesh patching technique is developed for the implementation of the GPRAMA method, which leads to an improved version of the popular GPRFS-ED method. The GPRAMA method can achieve better quality than the GPRFS-ED method but with lower computational cost.