arXiv subjects

Liuqiang Zhong

Publications and source records attributed to Liuqiang Zhong.

9 recordsLinked to original sources

arXiv2026

A correction adaptive two-grid finite element method for nonselfadjoint or indefinite elliptic problems

We propose, analyze, and numerically validate a correction adaptive two-grid finite element method (CAT-GFEM) for nonselfadjoint or indefinite elliptic problems. In contrast to the adaptive two-grid finite element method (ATGFEM) of Li and Zhang [SIAM J. Sci. Comput., 43 (2021), pp. A908-A928], which is restricted to symmetric positive-definite problems, the proposed method introduces an additional correction step that solves a small-scale discrete residual problem on the coarse mesh. This step entails negligible additional computational cost and allows us to show that the L2-norm error of the corrected discrete solution is a higher-order of the energy-norm error of the discrete solution. Using this result, we prove a contraction property for a suitable sum of quasi-errors on two successive adaptive meshes and establish convergence of the method. Numerical experiments illustrate the improved effectiveness and robustness of our method in comparison with ATGFEM.

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arXiv2025

A BDDC method with an adaptive coarse space for three-dimensional advection-diffusion problems

The solution of nonsymmetric but positive definite (NSPD) systems arising from advection-diffusion problems is an important research topic in science and engineering. Balancing domain decomposition by constraints with an adaptive coarse space(adaptive BDDC) constitute a significant class of nonoverlapping domain decomposition methods, commonly used for symmetric positive definite problems. In this paper, we propose an adaptive BDDC method that incorporates a class of edge generalized eigenvalue problems based on prior selected primal constraints to solve NSPD systems from advection-diffusion problems. Compared with the conventional adaptive BDDC method for such systems, the proposed approach further reduces the number of primal unknowns. Numerical experiments show that although the iteration count increases slightly, the overall computational time is significantly reduced.

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arXiv2023

Convergence and optimality of an adaptive modified weak Galerkin finite element method

An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this paper, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual-based a posteriori error estimator, an adaptive algorithm is proposed together with its convergence and quasi-optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix-Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results.

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arXiv2023

An iterative two-grid method for strongly nonlinear elliptic boundary value problems

We design and analyze an iterative two-grid algorithm for the finite element discretizations of strongly nonlinear elliptic boundary value problems in this paper. We propose an iterative two-grid algorithm, in which a nonlinear problem is first solved on the coarse space, and then a symmetric positive definite problem is solved on the fine space. The innovation of this paper lies in the establishment of a first convergence analysis, which requires simultaneous estimation of four interconnected error estimates. We also present some numerical experiments to confirm the efficiency of the proposed algorithm.

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arXiv2023

Iterative two-level algorithm for nonsymmetric or indefinite elliptic problems

In this paper, a new iterative two-level algorithm is presented for solving the finite element discretization for nonsymmetric or indefinite elliptic problems. The iterative two-level algorithm uses the same coarse space as the traditional two-grid algorithm, but its ``fine space'' uses the higher oder finite element space under the coarse grid. Therefore, the iterative two-level algorithm only needs one grid, and the computational cost is much lower than the traditional iterative two-grid algorithm. Finally, compared with the traditional two-grid algorithm, numerical experiments show that the computational cost is lower to achieve the same convergence order.

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arXiv2023

A Posterior Error Estimator for Mixed Interior Penalty Discontinuous Galerkin Finite Element Method for the H(curl)-Elliptic Problems

In this paper, we design the first residual type a posteriori error estimator for mixed interior penalty discontinuous Galerkin method for the H(curl)-elliptic problems. Then we prove that our residual based a posteriori error indicator is both reliable and efficient. At last, we present some numerical experiments to validate the performance of the indicator within an adaptive mesh refinement procedure.

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arXiv2022

A modified weak Galerkin method for $\boldsymbol{H}(\mathrm{curl})$-elliptic problem

In this paper, we design and analysis a modified weak Galerkin (MWG) finite element method for $\boldsymbol{H}(\mathrm{curl})-$elliptic problem. We first introduce a new discrete weak curl operator and the MWG finite element space. The modified weak Galerkin method does not require the penalty parameter by comparing with traditional DG methods. We prove optimal error estimates in energy norm. At last, we provide the numerical results to confirm these theoretical results.

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arXiv2021

Discontinuous Galerkin methods for semilinear elliptic boundary value problem

A discontinuous Galerkin (DG) scheme for solving semilinear elliptic problem is developed and analyzed in this paper. The DG finite element discretizations are established, and the corresponding existence and uniqueness theorem is proved by using Brouwer's fixed point method. Some optimal priori error estimates under both DG norm and $L^2$ norm are presented. Numerical results are also shown to confirm the efficiency of the proposed approach.

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arXiv2020

Adaptive-Multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems

In this paper, we are concerned with the weighted plane wave least-squares (PWLS) method for three-dimensional Helmholtz equations, and develop the multi-level adaptive BDDC algorithms for solving the resulting discrete system. In order to form the adaptive coarse components, the local generalized eigenvalue problems for each common face and each common edge are carefully designed. The condition number of the two-level adaptive BDDC preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of faces and edges per subdomain and the number of subdomains sharing a common edge. The efficiency of these algorithms is illustrated on a benchmark problem. The numerical results show the robustness of our two-level adaptive BDDC algorithms with respect to the wave number, the number of subdomains and the mesh size, and illustrate that our multi-level adaptive BDDC algorithm can reduce the scale of the coarse problem and can be used to solve large wave number problems efficiently.

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