arXiv subjects

Xuejun Xu

Publications and source records attributed to Xuejun Xu.

At least 37 records · Page 2Linked to original sources

arXiv2023

A family of stabilizer-free virtual elements on triangular meshes

A family of stabilizer-free $P_k$ virtual elements are constructed on triangular meshes. When choosing an accurate and proper interpolation, the stabilizer of the virtual elements can be dropped while the quasi-optimality is kept. The interpolating space here is the space of continuous $P_k$ polynomials on the Hsieh-Clough-Tocher macro-triangle, where the macro-triangle is defined by connecting three vertices of a triangle with its barycenter. We show that such an interpolation preserves $P_k$ polynomials locally and enforces the coerciveness of the resulting bilinear form. Consequently the stabilizer-free virtual element solutions converge at the optimal order. Numerical tests are provided to confirm the theory and to be compared with existing virtual elements.

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arXiv2023

Superconvergent P1 honeycomb virtual elements and lifted P3 solutions

When solving the Poisson equation on honeycomb hexagonal grids, we show that the $P_1$ virtual element is three-order superconvergent in $H^1$-norm, and two-order superconvergent in $L^2$ and $L^\infty$ norms. We define a local post-process which lifts the superconvergent $P_1$ solution to a $P_3$ solution of the optimal-order approximation. The theory is confirmed by a numerical test.

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arXiv2023

A non-overlapping optimization-based domain decomposition approach to component-based model reduction of incompressible flows

We present a component-based model order reduction procedure to efficiently and accurately solve parameterized incompressible flows governed by the Navier-Stokes equations. Our approach leverages a non-overlapping optimization-based domain decomposition technique to determine the control variable that minimizes jumps across the interfaces between sub-domains. To solve the resulting constrained optimization problem, we propose both Gauss-Newton and sequential quadratic programming methods, which effectively transform the constrained problem into an unconstrained formulation. Furthermore, we integrate model order reduction techniques into the optimization framework, to speed up computations. In particular, we incorporate localized training and adaptive enrichment to reduce the burden associated with the training of the local reduced-order models. Numerical results are presented to demonstrate the validity and effectiveness of the overall methodology.

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arXiv2023

Domain Decomposition Learning Methods for Solving Elliptic Problems

With recent advancements in computer hardware and software platforms, there has been a surge of interest in solving partial differential equations with deep learning-based methods, and the integration with domain decomposition strategies has attracted considerable attention owing to its enhanced representation and parallelization capacities of the network solution. While there are already several works that substitute the subproblem solver with neural networks for overlapping Schwarz methods, the non-overlapping counterpart has not been extensively explored because of the inaccurate flux estimation at interface that would propagate errors to neighbouring subdomains and eventually hinder the convergence of outer iterations. In this study, a novel learning approach for solving elliptic boundary value problems, i.e., the compensated deep Ritz method using neural network extension operators, is proposed to enable reliable flux transmission across subdomain interfaces, thereby allowing us to construct effective learning algorithms for realizing non-overlapping domain decomposition methods (DDMs) in the presence of erroneous interface conditions. Numerical experiments on a variety of elliptic problems, including regular and irregular interfaces, low and high dimensions, two and four subdomains, and smooth and high-contrast coefficients are carried out to validate the effectiveness of our proposed algorithms.

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arXiv2023

Dirichlet-Neumann learning algorithm for solving elliptic interface problems

Non-overlapping domain decomposition methods are natural for solving interface problems arising from various disciplines, however, the numerical simulation requires technical analysis and is often available only with the use of high-quality grids, thereby impeding their use in more complicated situations. To remove the burden of mesh generation and to effectively tackle with the interface jump conditions, a novel mesh-free scheme, i.e., Dirichlet-Neumann learning algorithm, is proposed in this work to solve the benchmark elliptic interface problem with high-contrast coefficients as well as irregular interfaces. By resorting to the variational principle, we carry out a rigorous error analysis to evaluate the discrepancy caused by the boundary penalty treatment for each decomposed subproblem, which paves the way for realizing the Dirichlet-Neumann algorithm using neural network extension operators. The effectiveness and robustness of our proposed methods are demonstrated experimentally through a series of elliptic interface problems, achieving better performance over other alternatives especially in the presence of erroneous flux prediction at interface.

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arXiv2023

On a Sharp Estimate of Overlapping Schwarz Methods in $\mathrm{H}(\mathrm{curl};Ω)$ and $\mathrm{H}(\mathrm{div};Ω)$

The previous proved-bound is $C(1+\frac{H^2}{δ^2})$ for the condition number of the overlapping domain decomposition $\mathrm{H}(\mathrm{curl};Ω)$ and $\mathrm{H}(\mathrm{div};Ω)$ methods, where $H$ and $δ$ are the sizes of subdomains and overlaps respectively. But all numerical results indicate that the best bound is $C(1+\frac{H}δ)$. In this work, we solve this long-standing open problem by proving that $C(1+\frac{H}δ)$ is indeed the best bound.

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arXiv2023

A Two-Level Block Preconditioned Jacobi-Davidson Method for Multiple and Clustered Eigenvalues of Elliptic Operators

In this paper, we propose a two-level block preconditioned Jacobi-Davidson (BPJD) method for efficiently solving discrete eigenvalue problems resulting from finite element approximations of $2m$th ($m = 1, 2$) order symmetric elliptic eigenvalue problems. Our method works effectively to compute the first several eigenpairs, including both multiple and clustered eigenvalues with corresponding eigenfunctions, particularly. The method is highly parallelizable by constructing a new and efficient preconditioner using an overlapping domain decomposition (DD). It only requires computing a couple of small scale parallel subproblems and a quite small scale eigenvalue problem per iteration. Our theoretical analysis reveals that the convergence rate of the method is bounded by $c(H)(1-C\frac{δ^{2m-1}}{H^{2m-1}})^{2}$, where $H$ is the diameter of subdomains and $δ$ is the overlapping size among subdomains. The constant $C$ is independent of the mesh size $h$ and the internal gaps among the target eigenvalues, demonstrating that our method is optimal and cluster robust. Meanwhile, the $H$-dependent constant $c(H)$ decreases monotonically to $1$, as $H \to 0$, which means that more subdomains lead to the better convergence rate. Numerical results supporting our theory are given.

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arXiv2022

Domain Decomposition Methods for Elliptic Problems with High Contrast Coefficients Revisited

In this paper, we revisit the nonoverlapping domain decomposition methods for solving elliptic problems with high contrast coefficients. Some interesting results are discovered. We find that the Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients. Actually, in the case of two subdomains, we show that their convergence rates are $O(ε)$, if $ν_1\llν_2$, where $ε= ν_1/ν_2$ and $ν_1,ν_2$ are coefficients of two subdomains. Moreover, in the case of many subdomains, the condition number bounds of Dirichlet-Neumann algorithm and Robin-Robin algorithm are $1+ε(1+\log(H/h))^2$ and $C+ε(1+\log(H/h))^2$, respectively, where $ε$ may be a very small number in the high contrast coefficients case. Besides, the convergence behaviours of the Neumann-Neumann algorithm and Dirichlet-Dirichlet algorithm may be independent of coefficients while they could not benefit from the discontinuous coefficients. Numerical experiments are preformed to confirm our theoretical findings.

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arXiv2021

A Two-Level Preconditioned Helmholtz-Jacobi-Davidson Method for the Maxwell Eigenvalue Problem

In this paper, based on a domain decomposition (DD) method, we shall propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space, we finally get the error reduction of this two-level PHJD method as $γ=c(H)(1-C\frac{δ^{2}}{H^{2}})$, where $C$ is a constant independent of the mesh size $h$ and the diameter of subdomains $H$, $δ$ is the overlapping size among the subdomains, and $c(H)$ decreasing as $H\to 0$, which means the greater the number of subdomains, the better the convergence rate. Numerical results supporting our theory shall be given.

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arXiv2020

Adaptive finite element approximation for steady-state Poisson-Nernst-Planck equations

In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a key contribution, the steady-state Poisson-Nernst-Planck equations are studied systematically and rigorous analysis for a residual-based a posteriori error estimate of the nonlinear system is presented. With the help of Schauder fixed point theorem, we show the solution existence and uniqueness of the linearized system derived by taking $G-$derivatives of the nonlinear system, followed by the proof of the relationship between the error of solution and the a posteriori error estimator $η$. Numerical experiments are given to validate the efficiency of the a posteriori error estimator and demonstrate the expected rate of convergence. In the further tests, adaptive mesh refinements for geometrical singularities and boundary layer effects are successfully observed.

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arXiv2018

Regularity results for the time-harmonic Maxwell equations with impedance boundary condition

This paper considers the time-harmonic Maxwell equations with impedance boundary condition.We present $H^2$-norm bound and other high-order norm bounds for strong solutions. The $H^2$-estimate have been derived in [M. Dauge, M. Costabel and S. Nicaise, Tech. Rep. 10-09, IRMAR (2010)] for the case with homogeneous boundary condition. Unfortunately, their method can not be applied to the inhomogeneous case. The main novelty of this paper is that we follow the spirit of the $H^1$-estimate in [R. Hiptmair, A. Moiola and I. Perugia, Math. Models Methods Appl. Sci., 21(2011), pp. 2263-2287] and modify the proof by applying two inequalities of Friedrichs' type to make the $H^1$-estimate move into $H^2$-estimate and $W^{m, p}$-estimate.Finally, the dependence of the regularity estimates on the wave number is obtained, which will play an important role in the convergence analysis of the numerical solutions for the time-harmonic Maxwell equations.

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arXiv2017

A Monotone Finite Volume Method for Time Fractional Fokker-Planck Equations

We develop a monotone finite volume method for the time fractional Fokker-Planck equations and theoretically prove its unconditional stability. We show that the convergence rate of this method is order 1 in space and if the space grid becomes sufficiently fine, the convergence rate can be improved to order 2. Numerical results are given to support our theoretical findings. One characteristic of our method is that it has monotone property such that it keeps the nonnegativity of some physical variables such as density, concentration, etc.

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arXiv2015

Domain Decomposition Methods for Space Fractional Partial Differential Equations

In this paper, a two-level additive Schwarz preconditioner is proposed for solving the algebraic systems resulting from the finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that the condition number of the preconditioned system is bounded by C(1+H/δ), where H is the maximum diameter of subdomains and δis the overlap size among the subdomains. Numerical results are given to support our theoretical findings.

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arXiv2015

Multigrid Methods for Space Fractional Partial Differential Equations

We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means the convergence rates of the methods are independent of the mesh size and mesh level. Moreover, our theoretical analysis and convergence results do not require regularity assumptions of the model problems. Numerical results are given to support our theoretical findings.

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arXiv2014

A FETI-DP preconditioner of discontinuous Galerkin method for multiscale problems in high constrast media

In this paper we consider second order elliptic partial differential equations with highly varying (heterogeneous) coefficients on a two-dimensional region. The problems are discretized by a composite finite element (FE) and discontinuous Galerkin (DG) Method. The fine grids are in general nonmatching across the subdomain boundaries, and the subdomain partitioning does not need to resolve the jumps in the coefficient. A FETI-DP preconditioner is proposed and analyzed to solve the resulting linear system. Numerical results are presented to support our theory.

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arXiv2013

A Robust Multilevel Method for Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation

A robust multilevel preconditioner based on the hybridizable discontinuous Galerkin method for the Helmholtz equation with high wave number is presented in this paper. There are two keys in our algorithm, one is how to choose a suitable intergrid transfer operator, and the other is using GMRES smoothing on coarse grids. The multilevel method is performed as a preconditioner in the outer GMRES iteration. To give a quantitative insight of our algorithm, we use local Fourier analysis to analyze the convergence property of the proposed multilevel method. Numerical results show that for fixed wave number, the convergence of the algorithm is mesh independent. Moreover, the performance of the algorithm depends relatively mildly on wave number.

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arXiv2013

Multilevel Preconditioner with Stable Coarse Grid Corrections for the Helmholtz Equation

In this paper we consider a class of robust multilevel precontioners for the Helmholtz equation with high wave number. The key idea in this work is to use the continuous interior penalty finite element methods (CIP-FEM) studied in \cite{Wu12,Wu12-hp} to construct the stable coarse grid correction problems. The multilevel methods, based on GMRES smoothing on coarse grids, are then served as a preconditioner in the outer GMRES iteration. In the one dimensional case, convergence property of the modified multilevel methods is analyzed by the local Fourier analysis. From our numerical results, we find that the proposed methods are efficient for a reasonable range of frequencies. The performance of the algorithms depends relatively mildly on wave number. In particular, only one GMRES smoothing step may guarantee the optimal convergence of our multilevel algorithm, which remedies the shortcoming of the multilevel algorithm in \cite{EEO01}.

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arXiv2012

A Hybridizable Discontinuous Galerkin Method for the Helmholtz Equation with High Wave Number

This paper analyzes the error estimates of the hybridizable discontinuous Galerkin (HDG) method for the Helmholtz equation with high wave number in two and three dimensions. The approximation piecewise polynomial spaces we deal with are of order $p\geq 1$. Through choosing a specific parameter and using the duality argument, it is proved that the HDG method is stable without any mesh constraint for any wave number $κ$. By exploiting the stability estimates, the dependence of convergence of the HDG method on $κ,h$ and $p$ is obtained. Numerical experiments are given to verify the theoretical results.

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