arXiv subjects

Haifeng Ji

Publications and source records attributed to Haifeng Ji.

10 recordsLinked to original sources

arXiv2026

Quadrilateral and Hexahedral Immersed Finite Element Methods for Elliptic Interface Problems

Immersed finite element (IFE) methods provide an effective framework for solving interface problems on unfitted meshes. The basic idea underlying IFE methods is to modify standard finite element spaces by enforcing interface conditions at certain points. It is known that, for nodal degrees of freedom, unisolvence of linear IFE basis functions on triangular elements is subject to a non-obtuse-angle condition for scalar diffusion coefficients. This limitation is more severe for tensor diffusion coefficients, for which the non-obtuse-angle condition is no longer sufficient for unisolvence. Even on rectangular meshes, conventional bilinear IFE basis functions may not be unisolvent for tensor diffusion coefficients. In this paper, we develop and analyze a nodal isoparametric IFE method on quadrilateral and hexahedral meshes and show that the unisolvence issue can be overcome by appropriately selecting the enforcement point of the discrete flux condition. The key observation is that, unlike triangular elements, the discrete flux in quadrilateral and hexahedral elements is not constant, which provides the flexibility to select such an enforcement point to ensure unisolvence. We provide a systematic procedure for this selection that not only ensures unisolvence of the IFE basis functions on general quadrilateral and hexahedral elements with either scalar or tensor-valued diffusion coefficients but also preserves the optimal approximation properties of the resulting IFE space. The proposed IFE method offers several advantages: flexibility for complex geometries, the absence of angle restrictions, and applicability to tensor diffusion coefficients, thereby overcoming the limitations of existing nodal IFE methods on rectangular and triangular meshes. Optimal error estimates are established and confirmed by numerical experiments.

math.NA↗

arXiv2026

A Pressure-Robust Nonconforming Immersed Finite Element Method for Stokes Interface Problems

It is well established that an appropriate modification of test functions may lead to pressure-robust mixed methods for Stokes problems. However, for immersed finite element approximations of Stokes interface problems on unfitted meshes, it remains unclear whether the velocity error is independent of the pressure, since the velocity and the pressure are coupled in one of the interface conditions. In this paper, we provide a positive answer through a novel decomposition of the discontinuous pressure into a continuous component and a velocity-dependent discontinuous component. We demonstrate that the immersed Crouzeix--Raviart/$P_0$ element method achieves pressure robustness via an $H(\operatorname{div})$-conforming reconstruction of the test functions on the right-hand side. The stability and optimal error estimates of the proposed method are established with constants independent of the interface position relative to the mesh. Numerical experiments are presented to validate the theoretical findings.

math.NA↗

arXiv2026

V2P-Manip: Learning Dexterous Manipulation from Monocular Human Videos

Achieving autonomous robotic dexterous manipulation requires precise, human-like action sequences at scale. As a scalable supplement to costly teleoperation data, extracting trajectories with both visual fidelity and physical plausibility from monocular videos represents a promising frontier in embodied AI. To this end, we introduce V2P-Manip, an efficient framework designed to learn dexterous manipulation policies directly from human demonstration videos. We establish an efficient, integrated pipeline encompassing 3D asset acquisition, trajectory estimation, and dexterous policy learning. To bridge the gap between visual perception and physical constraints, we introduce a two-stage refinement process to enforce spatial alignment and physical consistency. Evaluations on the TACO and OakInk benchmarks demonstrate that our approach significantly outperforms previous methods in pose accuracy, adaptability to unstructured environments, and training efficiency. Ultimately, experimental results confirm an average success rate of over 75% across multiple synthetic manipulation tasks and validate the adaptability of the extracted manipulation priors across diverse dexterous hand embodiments.

cs.RO↗

arXiv2025

An Immersed Finite Element Method for Anisotropic Elliptic Interface Problems with Nonhomogeneous Jump Conditions

A new finite element method (FEM) using meshes that do not necessarily align with the interface is developed for two- and three-dimensional anisotropic elliptic interface problems with nonhomogeneous jump conditions. The degrees of freedom of the proposed method are the same as those of traditional nonconforming FEMs, while the function space is modified to account for the jump conditions of the solution. The modified function space on an interface element is shown to exist uniquely, independent of the element's shape and the manner in which the interface intersects it. Optimal error estimates for the method, along with the usual bound on the condition number of the stiffness matrix, are proven, with the error constant independent of the interface's location relative to the mesh. To solve the resulting linear system, a preconditioner is proposed in which a Gauss-Seidel smoother with the interface correction is employed to ensure robustness against large jumps in the diffusion matrix. Numerical experiments are provided to demonstrate the optimal convergence of the proposed method and the efficiency of the preconditioner.

math.NA↗

arXiv2023

A Mini Immersed Finite Element Method for Two-Phase Stokes Problems on Cartesian Meshes

This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions, while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.

math.NA↗

arXiv2023

Analysis of nonconforming IFE methods and a new scheme for elliptic interface problems

In this paper, an important discovery has been found for nonconforming immersed finite element (IFE) methods using the integral values on edges as degrees of freedom for solving elliptic interface problems. We show that those IFE methods without penalties are not guaranteed to converge optimally if the tangential derivative of the exact solution and the jump of the coefficient are not zero on the interface. A nontrivial counter example is also provided to support our theoretical analysis. To recover the optimal convergence rates, we develop a new nonconforming IFE method with additional terms locally on interface edges. The new method is parameter-free which removes the limitation of the conventional partially penalized IFE method. We show the IFE basis functions are unisolvent on arbitrary triangles which is not considered in the literature. Furthermore, different from multipoint Taylor expansions, we derive the optimal approximation capabilities of both the Crouzeix-Raviart and the rotated-$Q_1$ IFE spaces via a unified approach which can handle the case of variable coefficients easily. Finally, optimal error estimates in both $H^1$- and $L^2$- norms are proved and confirmed with numerical experiments.

math.NA↗

arXiv2023

An immersed Crouzeix-Raviart finite element method in 2D and 3D based on discrete level set functions

This paper is devoted to the construction and analysis of immersed finite element (IFE) methods in three dimensions. Different from the 2D case, the points of intersection of the interface and the edges of a tetrahedron are usually not coplanar, which makes the extension of the original 2D IFE methods based on a piecewise linear approximation of the interface to the 3D case not straightforward. We address this coplanarity issue by an approach where the interface is approximated via discrete level set functions. This approach is very convenient from a computational point of view since in many practical applications the exact interface is often unknown, and only a discrete level set function is available. As this approach has also not be considered in the 2D IFE methods, in this paper we present a unified framework for both 2D and 3D cases. We consider an IFE method based on the traditional Crouzeix-Raviart element using integral values on faces as degrees of freedom. The novelty of the proposed IFE is the unisolvence of basis functions on arbitrary triangles/tetrahedrons without any angle restrictions even for anisotropic interface problems, which is advantageous over the IFE using nodal values as degrees of freedom. The optimal bounds for the IFE interpolation errors are proved on shape-regular triangulations. For the IFE method, optimal a priori error and condition number estimates are derived with constants independent of the location of the interface with respect to the unfitted mesh. The extension to anisotropic interface problems with tensor coefficients is also discussed. Numerical examples supporting the theoretical results are provided.

math.NA↗

arXiv2022

A new parameter free partially penalized immersed finite element and the optimal convergence analysis

This paper presents a new parameter free partially penalized immersed finite element method and convergence analysis for solving second order elliptic interface problems. A lifting operator is introduced on interface edges to ensure the coercivity of the method without requiring an ad-hoc stabilization parameter. The optimal approximation capabilities of the immersed finite element space is proved via a novel new approach that is much simpler than that in the literature. A new trace inequality which is necessary to prove the optimal convergence of immersed finite element methods is established on interface elements. Optimal error estimates are derived rigorously with the constant independent of the interface location relative to the mesh. The new method and analysis have also been extended to variable coefficients and three-dimensional problems. Numerical examples are also provided to confirm the theoretical analysis and efficiency of the new method.

math.NA↗

arXiv2021

An immersed $CR$-$P_0$ element for Stokes interface problems and the optimal convergence analysis

This paper presents and analyzes an immersed finite element (IFE) method for solving Stokes interface problems with a piecewise constant viscosity coefficient that has a jump across the interface. In the method, the triangulation does not need to fit the interface and the IFE spaces are constructed from the traditional $CR$-$P_0$ element with modifications near the interface according to the interface jump conditions. We prove that the IFE basis functions are unisolvent on arbitrary interface elements and the IFE spaces have the optimal approximation capabilities, although the proof is challenging due to the coupling of the velocity and the pressure. The stability and the optimal error estimates of the proposed IFE method are also derived rigorously. The constants in the error estimates are shown to be independent of the interface location relative to the triangulation. Numerical examples are provided to verify the theoretical results.

math.NA↗

arXiv2021

An immersed Raviart-Thomas mixed finite element method for elliptic interface problems on unfitted meshes

This paper presents a lowest-order immersed Raviart-Thomas mixed triangular finite element method for solving elliptic interface problems on unfitted meshes independent of the interface. In order to achieve the optimal convergence rates on unfitted meshes, an immersed finite element finite (IFE) is constructed by modifying the traditional Raviart-Thomas element. Some important properties are derived including the unisolvence of IFE basis functions, the optimal approximation capabilities of the IFE space and the corresponding commuting digram. Optimal error estimates are rigorously proved for the mixed IFE method and some numerical examples are also provided to validate the theoretical analysis.

math.NA↗