arXiv subjects

Qicheng Liu

Publications and source records attributed to Qicheng Liu.

8 recordsLinked to original sources

arXiv2026

Preconditioned Reconstructed Discontinuous Approximation For Elliptic Interface Problem on Unfitted Mesh

In this paper, we develop an efficient preconditioned unfitted finite element method for the elliptic interface problem, based on the reconstructed discontinuous approximation. The key idea is to impose suitable constraints on the local least squares reconstruction. These constraints ensure the stability near cut interface elements and, more importantly, establish a norm equivalence between the high-order space and the lowest-order piecewise constant space. This result allows us to construct an optimal preconditioner directly from the lowest-order system on the same unfitted mesh for any high-order scheme. The resulting method combines a cut discontinuous Galerkin formulation with Nitsche's penalty technique. The approximation space achieves arbitrarily high order accuracy with only one degree of freedom per element. We prove optimal error estimates and show that the condition number of the preconditioned system is uniformly bounded independently of the mesh size, coefficient contrast, and the location of the interface relative to the mesh. Multigrid algorithms are further designed to efficiently approximate the inverse of the lowest-order system matrix. Numerical experiments in two and three dimensions confirm the optimal convergence rates and demonstrate the robustness and efficiency of the proposed preconditioning method.

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arXiv2024

Preconditioned NonSymmetric/Symmetric Discontinuous Galerkin Method for Elliptic Problem with Reconstructed Discontinuous Approximation

In this paper, we propose and analyze an efficient preconditioning method for the elliptic problem based on the reconstructed discontinuous approximation method. We reconstruct a high-order piecewise polynomial space that arbitrary order can be achieved with one degree of freedom per element. This space can be directly used with the symmetric/nonsymmetric interior penalty discontinuous Galerkin method. Compared with the standard DG method, we can enjoy the advantage on the efficiency of the approximation. Besides, we establish an norm equivalence result between the reconstructed high-order space and the piecewise constant space. This property further allows us to construct an optimal preconditioner from the piecewise constant space. The upper bound of the condition number to the preconditioned symmetric/nonsymmetric system is shown to be independent of the mesh size. Numerical experiments are provided to demonstrate the validity of the theory and the efficiency of the proposed method.

math.NA↗

arXiv2024

A Finite Element Method by Patch Reconstruction for the Quad-Curl Problem Using Mixed Formulations

We develop a high order reconstructed discontinuous approximation (RDA) method for solving a mixed formulation of the quad-curl problem in two and three dimensions. This mixed formulation is established by adding an auxiliary variable to control the divergence of the field. The approximation space for the original variables is constructed by patch reconstruction with exactly one degree of freedom per element in each dimension and the auxiliary variable is approximated by the piecewise constant space. We prove the optimal convergence rate under the energy norm and also suboptimal $L^2$ convergence using a duality approach. Numerical results are provided to verify the theoretical analysis.

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arXiv2024

A Preconditioned Discontinuous Galerkin Method for Biharmonic Equation with $C^0$-Reconstructed Approximation

In this paper, we present a high-order finite element method based on a reconstructed approximation to the biharmonic equation. In our construction, the space is reconstructed from nodal values by solving a local least squares fitting problem per element. It is shown that the space can achieve an arbitrarily high-order accuracy and share the same nodal degrees of freedom with the $C^0$ linear space. The interior penalty discontinuous Galerkin scheme can be directly applied to the reconstructed space for solving the biharmonic equation. We prove that the numerical solution converges with optimal orders under error measurements. More importantly, we establish a norm equivalence between the reconstructed space and the continuous linear space. This property allows us to precondition the linear system arising from the high-order space by the linear space on the same mesh. This preconditioner is shown to be optimal in the sense that the condition number of the preconditioned system admits a uniform upper bound independent of the mesh size. Numerical examples in two and three dimensions are provided to illustrate the accuracy of the scheme and the efficiency of the preconditioning method.

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arXiv2024

An arbitrary order Reconstructed Discontinuous Approximation to Fourth-order Curl Problem

We present an arbitrary order discontinuous Galerkin finite element method for solving the fourth-order curl problem using a reconstructed discontinuous approximation method. It is based on an arbitrarily high-order approximation space with one unknown per element in each dimension. The discrete problem is based on the symmetric IPDG method. We prove a priori error estimates under the energy norm and the L^2 norm and show numerical results to verify the theoretical analysis.

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arXiv2023

An arbitrary order Reconstructed Discontinuous Approximation to Biharmonic Interface Problem

We present an arbitrary order discontinuous Galerkin finite element method for solving the biharmonic interface problem on the unfitted mesh. The approximation space is constructed by a patch reconstruction process with at most one degree freedom per element. The discrete problem is based on the symmetric interior penalty method and the jump conditions are weakly imposed by the Nitsche's technique. The C^2-smooth interface is allowed to intersect elements in a very general fashion and the stability near the interface is naturally ensured by the patch reconstruction. We prove the optimal a priori error estimate under the energy norm and the L^2 norm. Numerical results are provided to verify the theoretical analysis.

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arXiv2021

A discontinuous least squares finite element method for Helmholtz equations

We propose a discontinuous least squares finite element method for solving the Helmholtz equation. The method is based on the L2 norm least squares functional with the weak imposition of the continuity across the interior faces as well as the boundary conditions. We minimize the functional over the discontinuous polynomial spaces to seek numerical solutions. The wavenumber explicit error estimates to our method are established. The optimal convergence rate in the energy norm with respect to a fixed wavenumber is attained. The least squares functional can naturally serve as a posteriori estimator in the h-adaptive procedure. It is convenient to implement the code due to the usage of discontinuous elements. Numerical results in two and three dimensions are presented to verify the error estimates.

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arXiv2020

A discontinuous least squares finite element method for time-harmonic Maxwell equations

We propose and analyze a discontinuous least squares finite element method for solving the indefinite time-harmonic Maxwell equations. The scheme is based on the $L^2$ norm least squares functional with the weak imposition of the continuity across the interior faces. We minimize the functional over the piecewise polynomial spaces to seek numerical solutions. The method is shown to be stable without any constraint on the mesh size. We prove the convergence orders under both the energy norm and the $L^2$ norm. Numerical results in two and three dimensions are presented to verify the error estimates.

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