arXiv subjects

Quanling Deng

Publications and source records attributed to Quanling Deng.

At least 37 records · Page 2Linked to original sources

arXiv2022

Isogeometric Analysis of Bound States of a Quantum Three-Body Problem in 1D

In this paper, we initiate the study of isogeometric analysis (IGA) of a quantum three-body problem that has been well-known to be difficult to solve. In the IGA setting, we represent the wavefunctions by linear combinations of B-spline basis functions and solve the problem as a matrix eigenvalue problem. The eigenvalue gives the eigenstate energy while the eigenvector gives the coefficients of the B-splines that lead to the eigenstate. The major difficulty of isogeometric or other finite-element-method-based analyses lies in the lack of boundary conditions and a large number of degrees of freedom for accuracy. For a typical many-body problem with attractive interaction, there are bound and scattering states where bound states have negative eigenvalues. We focus on bound states and start with the analysis for a two-body problem. We demonstrate through various numerical experiments that IGA provides a promising technique to solve the three-body problem.

math.NA↗

arXiv2022

A boundary-penalized isogeometric analysis for second-order hyperbolic equations

Explicit time-marching schemes are popular for solving time-dependent partial differential equations; one of the biggest challenges these methods suffer is increasing the critical time-marching step size that guarantees numerical stability. In general, there are two ways to increase the critical step size. One is to reduce the stiffness of the spatially discretized system, while the other is to design time-marching schemes with larger stability regions. In this paper, we focus on the recently proposed explicit generalized-$α$ method for second-order hyperbolic equations and increase the critical step size by reducing the stiffness of the isogeometric-discretized system. In particular, we apply boundary penalization to lessen the system's stiffness. For $p$-th order $C^{p-1}$ isogeometric elements, we show numerically that the critical step size increases by a factor of $\sqrt{\frac{p^2-3p+6}{4}}$, which indicates the advantages of using the proposed method, especially for high-order elements. Various examples in one, two, and three dimensions validate the performance of the proposed technique.

math.NA↗

arXiv2022

Superfloe Parameterization with Physics Constraints for Uncertainty Quantification of Sea Ice Floes

The discrete element method (DEM) is providing a new modeling approach for describing sea ice dynamics. It exploits particle-based methods to characterize the physical quantities of each sea ice floe along its trajectory under Lagrangian coordinates. One major challenge in applying the DEM models is the heavy computational cost when the number of floes becomes large. In this paper, an efficient Lagrangian parameterization algorithm is developed, which aims at reducing the computational cost of simulating the DEM models while preserving the key features of the sea ice. The new parameterization takes advantage of a small number of artificial ice floes, named the superfloes, to effectively approximate a considerable number of the floes, where the parameterization scheme satisfies several important physics constraints. The physics constraints guarantee the superfloe parameterized system will have similar short-term dynamical behavior as the full system. These constraints also allow the superfloe parameterized system to accurately quantify the long-range uncertainty, especially the non-Gaussian statistical features, of the full system. In addition, the superfloe parameterization facilitates a systematic noise inflation strategy that significantly advances an ensemble-based data assimilation algorithm for recovering the unobserved ocean field underneath the sea ice. Such a new noise inflation method avoids ad hoc tunings as in many traditional algorithms and is computationally extremely efficient. Numerical experiments based on an idealized DEM model with multiscale features illustrate the success of the superfloe parameterization in quantifying the uncertainty and assimilating both the sea ice and the associated ocean field.

math.NA↗

arXiv2021

SoftFEM: revisiting the spectral finite element approximation of second-order elliptic operators

We propose, analyze mathematically, and study numerically a novel approach for the finite element approximation of the spectrum of second-order elliptic operators. The main idea is to reduce the stiffness of the problem by subtracting a least-squares penalty on the gradient jumps across the mesh interfaces from the standard stiffness bilinear form. This penalty bilinear form is similar to the known technique used to stabilize finite element approximations in various contexts. The penalty term is designed to dampen the high frequencies in the spectrum and so it is weighted here by a negative coefficient. The resulting approximation technique is called softFEM since it reduces the stiffness of the problem. The two key advantages of softFEM over the standard Galerkin FEM are to improve the approximation of the eigenvalues in the upper part of the discrete spectrum and to reduce the condition number of the stiffness matrix. We derive a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form. Then we prove that softFEM delivers the same optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. We next compare the discrete eigenvalues obtained when using Galerkin FEM and softFEM. Finally, a detailed analysis of linear softFEM for the 1D Laplace eigenvalue problem delivers a sensible choice for the softness parameter. With this choice, the stiffness reduction ratio scales linearly with the polynomial degree. Various numerical experiments illustrate the benefits of using softFEM over Galerkin FEM.

math.NA↗

arXiv2021

Outlier removal for isogeometric spectral approximation with the optimally-blended quadratures

It is well-known that outliers appear in the high-frequency region in the approximate spectrum of isogeometric analysis of the second-order elliptic operator. Recently, the outliers have been eliminated by a boundary penalty technique. The essential idea is to impose extra conditions arising from the differential equation at the domain boundary. In this paper, we extend the idea to remove outliers in the superconvergent approximate spectrum of isogeometric analysis with optimally-blended quadrature rules. We show numerically that the eigenvalue errors are of superconvergence rate $h^{2p+2}$ and the overall spectrum is outlier-free. The condition number and stiffness of the resulting algebraic system are reduced significantly. Various numerical examples demonstrate the performance of the proposed method.

math.NA↗

arXiv2021

A boundary penalization technique to remove outliers from isogeometric analysis on tensor-product meshes

We introduce a boundary penalization technique to improve the spectral approximation of isogeometric analysis (IGA). The technique removes the outliers appearing in the high-frequency region of the approximate spectrum when using the $C^{p-1}, p$-th ($p\ge3$) order isogeometric elements. We focus on the classical Laplacian (Dirichlet) eigenvalue problem in 1D to illustrate the idea and then use the tensor-product structure to generate the stiffness and mass matrices for multiple dimensional problems. To remove the outliers, we penalize the product of the higher-order derivatives from both the solution and test spaces at the domain boundary. Intuitively, we construct a better approximation by weakly imposing features of the exact solution. Effectively, we add terms to the variational formulation at the boundaries with minimal extra computational cost. We then generalize the idea to remove the outliers for the isogeometric analysis to the Neumann eigenvalue problem (for $p\ge2$). The boundary penalization does not change the test and solution spaces. In the limiting case when the penalty goes to infinity, we perform the dispersion analysis of $C^2$ cubic elements for Dirichlet eigenvalue problem and $C^1$ quadratic elements for Neumann eigenvalue problem. We obtain the analytical eigenpairs for the resulting matrix eigenvalue problems. Numerical experiments show optimal convergence rates for the eigenvalues and eigenfunctions of the discrete operator.

math.NA↗

arXiv2021

Analytical solutions to some generalized and polynomial eigenvalue problems

It is well-known that the finite difference discretization of the Laplacian eigenvalue problem $-Δu = λu$ leads to a matrix eigenvalue problem (EVP) $A x= λx$ where the matrix $A$ is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in Strang and MacNamara \cite{strang2014functions}. We generalize the results and develop analytical solutions to the generalized matrix eigenvalue problems (GEVPs) $A x= λBx$ which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.

math.NA↗

arXiv2021

Higher-order generalized-$α$ methods for parabolic problems

We propose a new class of high-order time-marching schemes with dissipation user-control and unconditional stability for parabolic equations. High-order time integrators can deliver the optimal performance of highly-accurate and robust spatial discretizations such as isogeometric analysis. The generalized-$α$ method delivers unconditional stability and second-order accuracy in time and controls the numerical dissipation in the discrete spectrum's high-frequency region. Our goal is to extend the generalized-$alpha$ methodology to obtain a high-order time marching methods with high accuracy and dissipation in the discrete high-frequency range. Furthermore, we maintain the stability region of the original, second-order generalized-$alpha$ method foe the new higher-order methods. That is, we increase the accuracy of the generalized-$α$ method while keeping the unconditional stability and user-control features on the high-frequency numerical dissipation. The methodology solve $k>1, k\in \mathbb{N}$ matrix problems and updates the system unknowns, which correspond to higher-order terms in Taylor expansions to obtain $(3/2k)^{th}$-order method for even $k$ and $(3/2k+1/2)^{th}$-order for odd $k$. A single parameter $ρ^\infty$ controls the dissipation, and the update procedure follows the formulation of the original second-order method. Additionally, we show that our method is A-stable and setting $ρ^\infty=0$ allows us to obtain an L-stable method. Lastly, we extend this strategy to analyze the accuracy order of a generic method.

math.NA↗

arXiv2021

On the Application of Stable Generalized Finite Element Method for Quasilinear Elliptic Two-Point BVP

In this paper, we discuss the application of the Generalized Finite Element Method (GFEM) to approximate the solutions of quasilinear elliptic equations with multiple interfaces in one dimensional space. The problem is characterized by spatial discontinuity of the elliptic coefficient that depends on the unknown solution. It is known that unless the partition of the domain matches the discontinuity configuration, accuracy of standard finite element techniques significantly deteriorates and standard refinement of the partition may not suffice. The GFEM is a viable alternative to overcome this predicament. It is based on the construction of certain enrichment functions supplied to the standard space that capture effects of the discontinuity. This approach is called stable (SGFEM) if it maintains an optimal rate of convergence and the conditioning of GFEM is not worse than that of the standard FEM. A convergence analysis is derived and performance of the method is illustrated by several numerical examples. Furthermore, it is known that typical global formulations such as FEMs do not enjoy the numerical local conservation property that is crucial in many conservation law-based applications. To remedy this issue, a Lagrange multiplier technique is adopted to enforce the local conservation. A numerical example is given to demonstrate the performance of proposed technique.

math.NA↗

arXiv2020

Isogeometric Residual Minimization Method (iGRM) with Direction Splitting Preconditoner for Stationary Advection-Diffusion Problems

In this paper, we introduce the isoGeometric Residual Minimization (iGRM) method. The method solves stationary advection-dominated diffusion problems. We stabilize the method via residual minimization. We discretize the problem using B-spline basis functions. We then seek to minimize the isogeometric residual over a spline space built on a tensor product mesh. We construct the solution over a smooth subspace of the residual. We can specify the solution subspace by reducing the polynomial order, by increasing the continuity, or by a combination of these. The Gramm matrix for the residual minimization method is approximated by a weighted H1 norm, which we can express as Kronecker products, due to the tensor-product structure of the approximations. We use the Gramm matrix as a preconditional which can be applied in a computational cost proportional to the number of degrees of freedom in 2D and 3D. Building on these approximations, we construct an iterative algorithm. We test the residual minimization method on several numerical examples, and we compare it to the Discontinuous Petrov-Galerkin (DPG) and the Streamline Upwind Petrov-Galerkin (SUPG) stabilization methods. The iGRM method delivers similar quality solutions as the DPG method, it uses smaller grids, it does not require breaking of the spaces, but it is limited to tensor-product meshes. The computational cost of the iGRM is higher than for SUPG, but it does not require the determination of problem specific parameters.

math.NA↗

arXiv2019

Split generalized-$α$ method: A linear-cost solver for a modified generalized-method for multi-dimensional second-order hyperbolic systems

We propose a variational splitting technique for the generalized-$α$ method to solve hyperbolic partial differential equations. We use tensor-product meshes to develop the splitting method, which has a computational cost that grows linearly with respect to the total number of degrees of freedom for multi-dimensional problems. We consider standard $C^0$ finite elements as well as smoother B-splines in isogeometric analysis for the spatial discretization. We also study the spectrum of the amplification matrix to establish the unconditional stability of the method. We then show that the stability behavior affects the overall behavior of the integrator on the entire interval and not only at the limits $0$ and $\infty$. We use various examples to demonstrate the performance of the method and the optimal approximation accuracy. For the numerical tests, we compute the $L_2$ and $H^1$ norms to show the optimal convergence of the discrete method in space and second-order accuracy in time.

math.NA↗

arXiv2019

Higher-order generalized-$α$ methods for hyperbolic problems

The generalized-$α$ time-marching method provides second-order accuracy in time and controls the numerical dissipation in the high-frequency region of the discrete spectrum. This method includes a wide range of time integrators. We increase the order of accuracy of the method while keeping the unconditional stability and the user-control on the high-frequency numerical dissipation. The dissipation is controlled by a single parameter as in the original method. Our high-order schemes require simple modifications of the available implementations of the generalized-$α$ method.

math.NA↗

arXiv2019

High-order generalized-$α$ methods

The generalized-$α$ method encompasses a wide range of time integrators. The method possesses high-frequency dissipation while minimizing unwanted low-frequency dissipation and the numerical dissipation can be controlled by the user. The method is unconditionally stable and is of second-order accuracy in time. We extend the second-order generalized-$α$ method to third-order in time while the numerical dissipation can be controlled in a similar fashion. We establish that the third-order method is unconditionally stable. We discuss a possible path to the generalization to higher order schemes. All these high-order schemes can be easily implemented into programs that already contain the second-order generalized-$α$ method.

math.NA↗

arXiv2019

Residual minimization for isogeometric analysis in reduced and mixed forms

Most variational forms of isogeometric analysis use highly-continuous basis functions for both trial and test spaces. For a partial differential equation with a smooth solution, isogeometric analysis with highly-continuous basis functions for trial space results in excellent discrete approximations of the solution. However, we observe that high continuity for test spaces is not necessary. In this work, we present a framework which uses highly-continuous B-splines for the trial spaces and basis functions with minimal regularity and possibly lower order polynomials for the test spaces. To realize this goal, we adopt the residual minimization methodology. We pose the problem in a mixed formulation, which results in a system governing both the solution and a Riesz representation of the residual. We present various variational formulations which are variationally-stable and verify their equivalence numerically via numerical tests.

math.NA↗

arXiv2018

A variationally separable splitting for the generalized-$α$ method for parabolic equations

We present a variationally separable splitting technique for the generalized-$α$ method for solving parabolic partial differential equations. We develop a technique for a tensor-product mesh which results in a solver with a linear cost with respect to the total number of degrees of freedom in the system for multi-dimensional problems. We consider finite elements and isogeometric analysis for the spatial discretization. The overall method maintains user-controlled high-frequency dissipation while minimizing unwanted low-frequency dissipation. The method has second-order accuracy in time and optimal rates ($h^{p+1}$ in $L^2$ norm and $h^p$ in $L^2$ norm of $\nabla u$) in space. We present the spectrum analysis on the amplification matrix to establish that the method is unconditionally stable. Various numerical examples illustrate the performance of the overall methodology and show the optimal approximation accuracy.

math.NA↗

arXiv2018

Higher order stable generalized finite element method for the elliptic eigenvalue problem with an interface in 1D

We study the generalized finite element methods (GFEMs) for the second-order elliptic eigenvalue problem with an interface in 1D. The linear stable generalized finite element methods (SGFEM) were recently developed for the elliptic source problem with interfaces. We first generalize SGFEM to arbitrary order elements and establish the optimal error convergence of the approximate solutions for the elliptic source problem with an interface. We then apply the abstract theory of spectral approximation of compact operators to establish the error estimation for the eigenvalue problem with an interface. The error estimations on eigenpairs strongly depend on the estimation of the discrete solution operator for the source problem. We verify our theoretical findings in various numerical examples including both source and eigenvalue problems.

math.NA↗

arXiv2018

Spectral approximation of elliptic operators by the Hybrid High-Order method

We study the approximation of the spectrum of a second-order elliptic differential operator by the Hybrid High-Order (HHO) method. The HHO method is formulated using cell and face unknowns which are polynomials of some degree $k\geq0$. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $h^{2t}$ and the eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size, $t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the eigenfunctions. Our theoretical findings, which improve recent error estimates for Hybridizable Discontinuous Galerkin (HDG) methods, are verified on various numerical examples including smooth and non-smooth eigenfunctions. Moreover, we observe numerically in one dimension for smooth eigenfunctions that the eigenvalues superconverge as $h^{2k+4}$ for a specific value of the stabilization parameter.

math.NA↗

arXiv2018

Dispersion optimized quadratures for isogeometric analysis

We develop and analyze quadrature blending schemes that minimize the dispersion error of isogeometric analysis up to polynomial order seven with maximum continuity in the span ($C^{p-1}$). The schemes yield two extra orders of convergence (superconvergence) on the eigenvalue errors, while the eigenfunction errors are of optimal convergence order. Both dispersion and spectrum analysis are unified in the form of a Taylor expansion for eigenvalue errors. As a consequence, the schemes increase the accuracy and robustness of isogeometric analysis for wave propagation as well as the differential eigenvalue problems. We analyze the methods' robustness and efficacy and utilize numerical examples to verify our analysis of the performance of the proposed schemes.

math.NA↗