arXiv subjects

Limin Ma

Publications and source records attributed to Limin Ma.

15 recordsLinked to original sources

arXiv2026

A Coupling Method of Mixed and Lagrange Finite Elements for Linear Elasticity Problem

This paper proposes a finite element method that couples mixed and Lagrange finite elements to efficiently capture stress concentrations in elasticity problems. The method employs conforming mixed finite elements in regions with stress concentration, while standard Lagrange elements are used elsewhere, achieving a balance between stress accuracy and computational efficiency. The well-posedness of the coupled formulation and optimal a priori error estimates are established, even when the size of the mixed finite element subregion is $O(h)$. Numerical experiments are presented to verify the theoretical convergence rates and to demonstrate the effectiveness and efficiency of the proposed method.

math.NA↗

arXiv2024

Evaluating LLMs for Text-to-SQL Generation With Complex SQL Workload

This study presents a comparative analysis of the a complex SQL benchmark, TPC-DS, with two existing text-to-SQL benchmarks, BIRD and Spider. Our findings reveal that TPC-DS queries exhibit a significantly higher level of structural complexity compared to the other two benchmarks. This underscores the need for more intricate benchmarks to simulate realistic scenarios effectively. To facilitate this comparison, we devised several measures of structural complexity and applied them across all three benchmarks. The results of this study can guide future research in the development of more sophisticated text-to-SQL benchmarks. We utilized 11 distinct Language Models (LLMs) to generate SQL queries based on the query descriptions provided by the TPC-DS benchmark. The prompt engineering process incorporated both the query description as outlined in the TPC-DS specification and the database schema of TPC-DS. Our findings indicate that the current state-of-the-art generative AI models fall short in generating accurate decision-making queries. We conducted a comparison of the generated queries with the TPC-DS gold standard queries using a series of fuzzy structure matching techniques based on query features. The results demonstrated that the accuracy of the generated queries is insufficient for practical real-world application.

cs.DB↗

arXiv2024

Supercloseness and asymptotic analysis of the Crouzeix-Raviart and enriched Crouzeix-Raviart elements for the Stokes problem

For the Crouzeix-Raviart and enriched Crouzeix-Raviart elements, asymptotic expansions of eigenvalues of the Stokes operator are derived by establishing two pseudostress interpolations, which admit a full one-order supercloseness with respect to the numerical velocity and the pressure, respectively. The design of these interpolations overcomes the difficulty caused by the lack of supercloseness of the canonical interpolations for the two nonconforming elements, and leads to an intrinsic and concise asymptotic analysis of numerical eigenvalues, which proves an optimal superconvergence of eigenvalues by the extrapolation algorithm. Meanwhile, an optimal superconvergence of postprocessed approximations for the Stokes equation is proved by use of this supercloseness. Finally, numerical experiments are tested to verify the theoretical results.

math.NA↗

arXiv2024

A direct finite element method for elliptic interface problems

In this paper, a direct finite element method is proposed for solving interface problems on unfitted meshes. This new method treats the two interface conditions as an $H^{\frac12}(\Gamma)\times H^{-\frac12}(\Gamma)$ pair for the mutual interaction across the interface, rather than the jumps of variables. A simple and straightforward finite element method is proposed based on this approach. This method solves the interface problem using conforming finite elements in one subdomain and conforming mixed finite elements in the other, with a natural integral term accounting for mutual interaction. Under reasonable assumptions, this direct finite element method is proved to be well-posed with an optimal a priori error analysis. Moreover, a simple lowest-order direct finite element method, using the linear element and the lowest-order Raviart-Thomas element, is analyzed to achieve the optimal a priori error estimate by verifying the aforementioned assumptions. Numerical tests are provided to confirm the theoretical results and the effectiveness of the direct finite element method.

math.NA↗

arXiv2022

An energy stable and maximum bound principle preserving scheme for the dynamic Ginzburg-Landau equations under the temporal gauge

This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg--Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Ned${\rm \acute{e}}$lec element and the linear element for spatial discretization, respectively; and a linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle (MBP) of the order parameter and the energy dissipation law in the discrete sense are proved. The discrete energy stability and MBP-preservation can guarantee the stability and validity of the numerical simulations, and further facilitate the adoption of an adaptive time-stepping strategy, which often plays an important role in long-time simulations of vortex dynamics, especially when the applied magnetic field is strong. An optimal error estimate of the proposed scheme is also given. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field.

math.NA↗

arXiv2022

An efficient iterative method for dynamical Ginzburg-Landau equations

In this paper, we propose a new finite element approach to simulate the time-dependent Ginzburg-Landau equations under the temporal gauge, and design an efficient preconditioner for the Newton iteration of the resulting discrete system. The new approach solves the magnetic potential in H(curl) space by the lowest order of the second kind Nedelec element. This approach offers a simple way to deal with the boundary condition, and leads to a stable and reliable performance when dealing with the superconductor with reentrant corners. The comparison in numerical simulations verifies the efficiency of the proposed preconditioner, which can significantly speed up the simulation in large-scale computations.

math.NA↗

arXiv2022

Incremental Information Gain Mining Of Temporal Relational Streams

This paper studies the problem of mining for data values with high information gain in relational tables. High information gain can help data analysts and secondary data mining algorithms gain insights into strong statistical dependencies and causality relationship between key metrics. In this paper, we will study the problem of high information gain identification for scenarios involving temporal relations where new records are added continuously to the relations. We show that information gain can be efficiently maintained in an incremental fashion, making it possible to monitor continuously high information gain values.

cs.IR↗

arXiv2022

New Fourth Order Postprocessing Techniques for Plate Bending Eigenvalues by Morley Element

In this paper, we propose and analyze the extrapolation method and asymptotically exact a posterior error estimate for eigenvalues of the Morley element. We establish an asymptotic expansion of eigenvalues, and prove an optimal result for this expansion and the corresponding extrapolation method. We also design an asymptotically exact a posterior error estimate and propose new approximate eigenvalues with higher accuracy by utilizing this a posteriori error estimate. Finally, several numerical experiments are considered to confirm the theoretical results and compare the performance of the proposed methods.

math.NA↗

arXiv2021

New Discontinuous Galerkin Algorithms and Analysis for Linear Elasticity with Symmetric Stress Tensor

This paper presents a new and unified approach to the derivation and analysis of many existing, as well as new discontinuous Galerkin methods for linear elasticity problems. The analysis is based on a unified discrete formulation for the linear elasticity problem consisting of four discretization variables: strong symmetric stress tensor $\dsig$ and displacement $\du$ inside each element, and the modifications of these two variables $\hsig$ and $\hu$ on elementary boundaries of elements. Motivated by many relevant methods in the literature, this formulation can be used to derive most existing discontinuous, nonconforming and conforming Galerkin methods for linear elasticity problems and especially to develop a number of new discontinuous Galerkin methods. Many special cases of this four-field formulation are proved to be hybridizable and can be reduced to some known hybridizable discontinuous Galerkin, weak Galerkin and local discontinuous Galerkin methods by eliminating one or two of the four fields. As certain stabilization parameter tends to zero, this four-field formulation is proved to converge to some conforming and nonconforming mixed methods for linear elasticity problems. Two families of inf-sup conditions, one known as $H^1$-based and the other known as $H({\rm div})$-based, are proved to be uniformly valid with respect to different choices of discrete spaces and parameters. These inf-sup conditions guarantee the well-posedness of the new proposed methods and also offer a new and unified analysis for many existing methods in the literature as a by-product. Some numerical examples are provided to verify the theoretical analysis including the optimal convergence of the new proposed methods.

math.NA↗

arXiv2021

Superconvergence of Discontinuous Galerkin methods for Elliptic Boundary Value Problems

In this paper, we present a unified analysis of the superconvergence property for a large class of mixed discontinuous Galerkin methods. This analysis applies to both the Poisson equation and linear elasticity problems with symmetric stress formulations. Based on this result, some locally postprocess schemes are employed to improve the accuracy of displacement by order min(k+1, 2) if polynomials of degree k are employed for displacement. Some numerical experiments are carried out to validate the theoretical results.

math.NA↗

arXiv2020

Asymptotic expansions of eigenvalues by both the Crouzeix-Raviart and enriched Crouzeix-Raviart elements

Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix--Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart--Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart--Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions.

math.NA↗

arXiv2019

Asymptotically exact a posteriori error estimates of eigenvalues by the Crouzeix-Raviart element and enriched Crouzeix-Raviart element

Two asymptotically exact a posteriori error estimates are proposed for eigenvalues by the nonconforming Crouzeix--Raviart and enriched Crouzeix-- Raviart elements. The main challenge in the design of such error estimators comes from the nonconformity of the finite element spaces used. Such nonconformity causes two difficulties, the first one is the construction of high accuracy gradient recovery algorithms, the second one is a computable high accuracy approximation of a consistency error term. The first difficulty was solved for both nonconforming elements in a previous paper. Two methods are proposed to solve the second difficulty in the present paper. In particular, this allows the use of high accuracy gradient recovery techniques. Further, a post-processing algorithm is designed by utilizing asymptotically exact a posteriori error estimators to construct the weights of a combination of two approximate eigenvalues. This algorithm requires to solve only one eigenvalue problem and admits high accuracy eigenvalue approximations both theoretically and numerically.

math.NA↗

arXiv2019

Optimal Superconvergence Analysis for the Crouzeix-Raviart and the Morley elements

In this paper, an improved superconvergence analysis is presented for both the Crouzeix-Raviart element and the Morley element. The main idea of the analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution for the first order mixed Raviart--Thomas element and the mixed Hellan--Herrmann--Johnson element, respectively. This, in particular, allows for proving a full one order superconvergence result for these two mixed finite elements. Finally, a full one order superconvergence result of both the Crouzeix-Raviart element and the Morley element follows from their special relations with the first order mixed Raviart--Thomas element and the mixed Hellan--Herrmann--Johnson element respectively. Those superconvergence results are also extended to mildly-structured meshes.

math.NA↗

arXiv2018

High accuracy methods for eigenvalues of elliptic operators by nonconforming elements

In this paper, three high-accuracy methods for eigenvalues of second order elliptic operators are proposed by using the nonconforming Crouzeix-Raviart(CR for short) element and the nonconforming enriched Crouzeix-Raviart(ECR for short) element. They are based on a crucial full one order superconvergence of the first order mixed Raviart-Thomas(RT for short) element. The main ingredient of such a superconvergence analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution of the RT element. In particular, it allows for some vital cancellation between terms in one key sum of boundary terms. Consequently, a full one order superconvergence follows from a special relation between the CR element and the RT element, and the equivalence between the ECR element and the RT element for these two nonconforming elements. These superconvergence results improve those in literature from a half order to a full one order for the RT element, the CR element and the ECR element. Based on the aforementioned superconvergence of the RT element, asymptotic expansions of eigenvalues are established and employed to achieve high accuracy extrapolation methods for these two nonconforming elements. In contrast to a classic analysis in literature, the novelty herein is to use not only the canonical interpolations of these nonconforming elements but also that of the RT element to analyze such asymptotic expansions. Based on the superconvergence of these nonconforming elements, asymptotically exact a posteriori error estimators of eigenvalues are constructed and analyzed for them. Finally, two post-processing methods are proposed to improve accuracy of approximate eigenvalues by employing these a posteriori error estimators.Numerical tests are provided to justify and compare the performance of the aforementioned methods.

math.NA↗

arXiv2016

A Penalized Crouzeix-Raviart Element Method for Second Order Elliptic Eigenvalue Problems

In this paper we propose a penalized Crouzeix-Raviart element method for eigenvalue problems of second order elliptic operators. The key idea is to add a penalty term to tune the local approximation property and the global continuity property of the discrete eigenfunctions. The feature of this method is that by adjusting the penalty parameter, the resulted discrete eigenvalues can be in a state of "chaos", and consequently a large portion of them can be reliable and approximate the exact ones with high accuracy. Furthermore, we design an algorithm to select such a quasi-optimal penalty parameter. Finally, we provide numerical tests to demonstrate the performance of the proposed method.

math.NA↗