arXiv subjects

Chupeng Ma

Publications and source records attributed to Chupeng Ma.

18 recordsLinked to original sources

arXiv2025

Multiscale model reduction and two-level Schwarz preconditioner for H(curl) elliptic problems

This paper addresses the efficient solution of linear systems arising from curl-conforming finite element discretizations of $H(\mathrm{curl})$ elliptic problems with heterogeneous coefficients. We first employ the discrete form of a multiscale spectral generalized finite element method (MS-GFEM) for model reduction and prove that the method exhibits exponential convergence with respect to the number of local degrees of freedom. The proposed method and its convergence analysis are applicable in broad settings, including general heterogeneous ($L^{\infty}$) coefficients, domains and subdomains with nontrivial topology, irregular subdomain geometries, and high-order finite element discretizations. Furthermore, we formulate the method as an iterative solver, yielding a two-level restricted additive Schwarz type preconditioner based on the MS-GFEM coarse space. The GMRES algorithm, applied to the preconditioned system, is shown to converge at a rate of at least $\Lambda$, where $\Lambda$ denotes the error bound of the discrete MS-GFEM approximation. Numerical experiments in both two and three dimensions demonstrate the superior performance of the proposed methods in terms of dimensionality reduction.

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arXiv2024

Two-level Restricted Additive Schwarz preconditioner based on Multiscale Spectral Generalized FEM for Heterogeneous Helmholtz Problems

We present and analyze a two-level restricted additive Schwarz (RAS) preconditioner for heterogeneous Helmholtz problems, based on a multiscale spectral generalized finite element method (MS-GFEM) proposed in [C. Ma, C. Alber, and R. Scheichl, SIAM. J. Numer. Anal., 61 (2023), pp. 1546--1584]. The preconditioner uses local solves with impedance boundary conditions, and a global coarse solve based on the MS-GFEM approximation space constructed from local eigenproblems. It is derived by first formulating MS-GFEM as a Richardson iterative method, and without using an oversampling technique, reduces to the preconditioner recently proposed and analyzed in [Q. Hu and Z.Li, arXiv 2402.06905]. We prove that both the Richardson iterative method and the preconditioner used within GMRES converge at a rate of $\Lambda$ under some reasonable conditions, where $\Lambda$ denotes the error of the underlying MS-GFEM \rs{approximation}. Notably, the convergence proof of GMRES does not rely on the `Elman theory'. An exponential convergence property of MS-GFEM, resulting from oversampling, ensures that only a few iterations are needed for convergence with a small coarse space. Moreover, the convergence rate $\Lambda$ is not only independent of the fine-mesh size $h$ and the number of subdomains, but decays with increasing wavenumber $k$. In particular, in the constant-coefficient case, with $h\sim k^{-1-\gamma}$ for some $\gamma\in (0,1]$, it holds that $\Lambda \sim k^{-1+\frac{\gamma}{2}}$. We present extensive numerical experiments to illustrate the performance of the preconditioner, including 2D and 3D benchmark geophysics tests, and a high-contrast coefficient example arising in applications.

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arXiv2024

Fast-convergent two-level restricted additive Schwarz methods based on optimal local approximation spaces

This paper proposes a two-level restricted additive Schwarz (RAS) method for multiscale PDEs, built on top of a multiscale spectral generalized finite element method (MS-GFEM). The method uses coarse spaces constructed from optimal local approximation spaces, which are based on local eigenproblems posed on (discrete) harmonic spaces. We rigorously prove that the method, used as an iterative solver or as a preconditioner for GMRES, converges at a rate of $\Lambda$, where $\Lambda$ represents the error of the underlying MS-GFEM. The exponential convergence property of MS-GFEM, which is indepdendent of the fine mesh size $h$ even for highly oscillatory and high contrast coefficients, thus guarantees convergence in a few iterations with a small coarse space. We develop the theory in an abstract framework, and demonstrate its generality by applying it to various elliptic problems with highly heterogeneous coefficients, including $H({\rm curl})$ elliptic problems. The performance of the proposed method is systematically evaluated and illustrated via applications to two and three dimensional heterogeneous PDEs, including challenging elasticity problems in realistic composite aero-structures.

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arXiv2024

A Mixed Multiscale Spectral Generalized Finite Element Method

We present a multiscale mixed finite element method for solving second order elliptic equations with general $L^{\infty}$-coefficients arising from flow in highly heterogeneous porous media. Our approach is based on a multiscale spectral generalized finite element method (MS-GFEM) and exploits the superior local mass conservation properties of mixed finite elements. Following the MS-GFEM framework, optimal local approximation spaces are built for the velocity field by solving local eigenvalue problems over generalized harmonic spaces. The resulting global velocity space is then enriched suitably to ensure inf-sup stability. We develop the mixed MS-GFEM for both continuous and discrete formulations, with Raviart-Thomas based mixed finite elements underlying the discrete method. Exponential convergence with respect to local degrees of freedom is proven at both the continuous and discrete levels. Numerical results are presented to support the theory and to validate the proposed method.

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arXiv2023

A unified framework for multiscale spectral generalized FEMs and low-rank approximations to multiscale PDEs

This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.

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arXiv2023

Scalable multiscale-spectral GFEM with an application to composite aero-structures

In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. The crucial novelty lies in the introduction of A-harmonicity in the local approximation spaces, which in contrast to [Babuska, Lipton, Multiscale Model. Simul. 9, 2011] is enforced more efficiently via a constraint in the local eigenproblems. This significant modification leads to excellent approximation properties, which turn out to be essential to capture accurately material strains and stresses with a low dimensional approximation space, hence maximising model order reduction. The implementation of the framework in the DUNE software package, as well as a detailed description of all components of the method are presented and exemplified on a composite laminated beam under compressive loading. The excellent parallel scalability of the method, as well as its superior performance compared to the related, previously introduced GenEO method are demonstrated on two realistic application cases, including a C-shaped wing spar with complex geometry. Further, by allowing low-cost approximate solves for closely related models or geometries this efficient, novel technology provides the basis for future applications in optimisation or uncertainty quantification on challenging problems in composite aero-structures.

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arXiv2022

Wavenumber explicit convergence of a multiscale generalized finite element method for heterogeneous Helmholtz problems

In this paper, a generalized finite element method (GFEM) with optimal local approximation spaces for solving high-frequency heterogeneous Helmholtz problems is systematically studied. The local spaces are built from selected eigenvectors of carefully designed local eigenvalue problems defined on generalized harmonic spaces. At both continuous and discrete levels, $(i)$ wavenumber explicit and nearly exponential decay rates for local and global approximation errors are obtained without any assumption on the size of subdomains; $(ii)$ a quasi-optimal convergence of the method is established by assuming that the size of subdomains is $O(1/k)$ ($k$ is the wavenumber). A novel resonance effect between the wavenumber and the dimension of local spaces on the decay of error with respect to the oversampling size is implied by the analysis. Furthermore, for fixed dimensions of local spaces, the discrete local errors are proved to converge as $h\rightarrow 0$ ($h$ denoting the mesh size) towards the continuous local errors. The method at the continuous level extends the plane wave partition of unity method [I. Babuska and J. M. Melenk, Int.\;J.\;Numer.\;Methods Eng., 40 (1997), pp.~727--758] to the heterogeneous-coefficients case, and at the discrete level, it delivers an efficient non-iterative domain decomposition method for solving discrete Helmholtz problems resulting from standard FE discretizations. Numerical results are provided to confirm the theoretical analysis and to validate the proposed method.

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arXiv2022

Exponential convergence of a generalized FEM for heterogeneous reaction-diffusion equations

A generalized finite element method is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter $\varepsilon$, based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size $\delta^{\ast}$. The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard FE discretizations. Exponential decay rates for local approximation errors with respect to $\delta^{\ast}/\varepsilon$ and $\delta^{\ast}/h$ (at the discrete level with $h$ denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to $\varepsilon$ in the standard $H^{1}$ norm, and that if the oversampling size is relatively large with respect to $\varepsilon$ and $h$ (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results.

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arXiv2022

Overlapping Schwarz methods with GenEO coarse spaces for indefinite and non-self-adjoint problems

GenEO (`Generalised Eigenvalue problems on the Overlap') is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers which are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection-diffusion-reaction problems, which may be non-self-adjoint and indefinite, and whose discretisations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalised eigenvalue problem based on a self-adjoint and positive-definite subproblem. We prove estimates on GMRES iteration counts which are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on variations of the other coefficients. These are proved under the assumption that the subdomain diameter is sufficiently small and the eigenvalue tolerance for building the coarse space is sufficiently large. While the iteration number estimates do grow as the non-self-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus we obtain an iterative solver which is efficient in parallel and very effective for a wide range of convection--diffusion--reaction problems.

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arXiv2021

Novel design and analysis of generalized FE methods based on locally optimal spectral approximations

In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a partition of unity are presented. These new spaces have advantages over those proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. First, in addition to a nearly exponential decay rate of the local approximation errors with respect to the dimensions of the local spaces, the rate of convergence with respect to the size of the oversampling region is also established. Second, the theoretical results hold for problems with mixed boundary conditions defined on general Lipschitz domains. Finally, an efficient and easy-to-implement technique for generating the discrete $A$-harmonic spaces is proposed which relies on solving an eigenvalue problem associated with the Dirichlet-to-Neumann operator, leading to a substantial reduction in computational cost. Numerical experiments are presented to support the theoretical analysis and to confirm the effectiveness of the new method.

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arXiv2021

Error estimates for fully discrete generalized FEMs with locally optimal spectral approximations

This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation spaces are constructed using eigenvectors of local eigenvalue problems solved by the finite element method on some sufficiently fine mesh with mesh size $h$. The error bound of the discrete GFEM approximation is proved to converge as $h\rightarrow 0$ towards that of the continuous GFEM approximation, which was shown to decay nearly exponentially in previous works. Moreover, even for fixed mesh size $h$, a nearly exponential rate of convergence of the local approximation errors with respect to the dimension of the local spaces is established. An efficient and accurate method for solving the discrete eigenvalue problems is proposed by incorporating the discrete $A$-harmonic constraint directly into the eigensolver. Numerical experiments are carried out to confirm the theoretical results and to demonstrate the effectiveness of the method.

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arXiv2021

GenEO coarse spaces for heterogeneous indefinite elliptic problems

Motivated by recent work on coarse spaces for Helmholtz problems, we provide in this paper a comparative study on the use of spectral coarse spaces of GenEO type for heterogeneous indefinite elliptic problems within an additive overlapping Schwarz method. In particular, we focus here on two different but related formulations of local generalised eigenvalue problems and compare their performance numerically. Even though their behaviour seems to be very similar for several well-known heterogeneous test cases that are mildly indefinite, only one of the coarse spaces has so far been analysed theoretically, while the other one leads to a significantly more robust domain decomposition method when the indefiniteness is increased. We present a summary of upcoming results developing such a theory and describe how the numerical experiments illustrate it.

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arXiv2020

Efficient multiscale algorithms for simulating nonlocal optical response of metallic nanostructure arrays

In this paper, we consider numerical simulations of the nonlocal optical response of metallic nanostructure arrays inside a dielectric host, which is of particular interest to the nanoplasmonics community due to many unusual properties and potential applications. Mathematically, it is described by Maxwell's equations with discontinuous coefficients coupled with a set of Helmholtz-type equations defined only on the domains of metallic nanostructures. To solve this challenging problem, we develop an efficient multiscale method consisting of three steps. First, we extend the system into the domain occupied by the dielectric medium in a novel way and result in a coupled system with rapidly oscillating coefficients. A rigorous analysis of the error between the solutions of the original system and the extended system is given. Second, we derive the homogenized system and define the multiscale approximate solution for the extended system by using the multiscale asymptotic method. Third, to fix the inaccuracy of the multiscale asymptotic method inside the metallic nanostructures, we solve the original system in each metallic nanostructure separately with boundary conditions given by the multiscale approximate solution. A fast algorithm based on the $LU$ decomposition is proposed for solving the resulting linear systems. By applying the multiscale method, we obtain the results that are in good agreement with those obtained by solving the original system directly at a much lower computational cost. Numerical examples are provided to validate the efficiency and accuracy of the proposed method.

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arXiv2019

Mathematical and numerical analysis of a nonlocal Drude model in nanoplasmonics

In this paper, we consider the frequency-domain Maxwell's equations coupled to a nonlocal Drude model which describes the nonlocal optical response in metallic nanostructures. We prove the existence and uniqueness of weak solutions to the coupled equations. A Galerkin finite element method based on the Raviart--Thomas and Nédélec elements is proposed to solve the equations and the associated error estimates are given. This is the first work on the mathematical and numerical analysis of this model. Numerical examples are presented to verify our theoretical analysis.

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arXiv2017

Mathematical and numerical analysis of the time-dependent Maxwell--Schr\"{o}dinger Equations in the Coulomb gauge

In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell--Schr\"{o}dinger equations in the Coulomb gauge. We first prove the global existence of weak solutions to the equations. Next we propose an energy-conserving fully discrete finite element scheme for the system and prove the existence and uniqueness of solutions to the discrete system. The optimal error estimates for the numerical scheme without any time-step restrictions are then derived. Numerical results are provided to support our theoretical analysis.

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arXiv2017

Error estimates of the Crank-Nicolson Galerkin method for the time-dependent Maxwell-Schr\"{o}dinger equations under the Lorentz gauge

In this paper we study the numerical method and the convergence for solving the time-dependent Maxwell-Schr\"{o}dinger equations under the Lorentz gauge. An alternating Crank-Nicolson finite element method for solving the problem is presented and the optimal error estimate for the numerical algorithm is obtained by a mathematical inductive method. Numerical examples are then carried out to confirm the theoretical results.

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arXiv2017

A Crank-Nicolson Finite Element Method and the Optimal Error Estimates for the modified Time-dependent Maxwell-Schr\"{o}dinger Equations

In this paper we consider the initial-boundary value problem for the time-dependent Maxwell-Schr\"{o}dinger equations, which arises in the interaction between the matter and the electromagnetic field for the semiconductor quantum devices. A Crank-Nicolson finite element method for solving the problem is presented. The optimal energy-norm error estimates for the numerical algorithm without any time-step restrictions are derived. Numerical tests are then carried out to confirm the theoretical results.

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arXiv2017

Multiscale Approach and The Convergence for the time-dependent Maxwell-Schr\"{o}dinger System in Heterogeneous Nanostructures

This paper discusses the multiscale approach and the convergence of the time-dependent Maxwell-Schr\"{o}dinger system with rapidly oscillating discontinuous coefficients arising from the modeling of a heterogeneous nanostructure with a periodic microstructure. The homogenization method and the multiscale asymptotic method for the nonlinear coupled equations are presented. The efficient numerical algorithms based on the above methods are proposed. Numerical simulations are then carried out to validate the method presented in this paper.

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