Limin Ma

Jun Hu, Limin Ma

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.

Limin Ma, Ken Pu, Ying Zhu

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.

Wei Chen, Jun Hu, Limin Ma et al.

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.

Jun Hu, Limin Ma

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.