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In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous Dirichlet ...

In this paper, we propose a type of tensor-neural-network-based machine learning method to compute multi-eigenpairs of high dimensional eigenvalue problems without Monte-Carlo procedure. Solving multi-eigenvalues and their corresponding ...

In this paper, some enhanced error estimates are derived for the augmented subspace methods which are designed for solving eigenvalue problems. For the first time, we strictly prove that the augmented subspace methods have the second order ...

We design a multilevel correction type of adaptive finite element method based on the moving mesh technique for solving nonlinear eigenvalue problems. In this paper, we take the ground state of Bose–Einstein condensates as the example of a ...

In this paper, we present a nonnested augmented subspace algorithm and its multilevel correction method for solving elliptic eigenvalue problems with curved interfaces. The augmented subspace algorithm and the corresponding multilevel correction ...

Based on the numerical method proposed in Hu et al. (2018) [22] for Kohn-Sham equation, further improvement on the efficiency is obtained in this paper by i). designing a numerical method with the strategy of separately handling the ...

• An accelerating multilevel correction AFEM is designed to solve Kohn-Sham equation.

Many nonlocal models have adopted Euclidean balls as the nonlocal interaction neighborhoods. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such balls. A crucial question is to what extent such ...

In this paper, a new kind of multigrid method is proposed for the ground state solution of Bose–Einstein condensates based on Newton iteration scheme. Instead of treating eigenvalue $\lambda$ and eigenvector u separately, we regard the eigenpair $(\lambda ,u)$ as ...$\mathbb{}{}_{\mathrm{}}^{\mathrm{}}$

A type of parallel augmented subspace scheme for eigenvalue problems is proposed by using coarse space in the multigrid method. With the help of coarse space in the multigrid method, solving the eigenvalue problem in the finest space is decomposed into ...

In this paper, a cascadic multigrid method is proposed to solve nonsymmetric eigenvalue problems. Based on the multilevel correction method, the proposed method transforms a nonsymmetric eigenvalue problem solving on the finest finite ...

In this paper, we propose a computable error estimate of the Gross–Pitaevskii equation for the ground state solution of the Bose–Einstein condensate by the general conforming finite element method on general meshes. Based on this error estimate, ...

Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space ...

We present a method for the fast computation of the eigenpairs of a bijective positive symmetric linear operator $\mathcal{L}$. The method is based on a combination of operator adapted wavelets (gamblets) with hierarchical subspace correction. First, ...

To provide mathematically rigorous eigenvalue bounds for the Steklov eigenvalue problem, an enhanced version of the eigenvalue estimation algorithm developed by the third author is proposed. Compared with the existing algorithm, which deals with ...

We propose a numerical strategy to generate a sequence of anisotropic meshes and select appropriate stabilization parameters simultaneously for linear SUPG method solving two dimensional convection-dominated convection---diffusion equations. Since the ...

An adaptive finite element method for eigenvalue problems is proposed based on the multilevel correction scheme. Different from the standard adaptive finite element method which requires solving eigenvalue problems on adaptively refined triangulations, ...

In this paper, we give a numerical analysis for the transmission eigenvalue problem by the finite element method. A type of multilevel correction method is proposed to solve the transmission eigenvalue problem. The multilevel correction method can ...

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which ...

In this paper, a full (nested) multigrid scheme is proposed to solve eigenvalue problems. The idea here is to use a correction method to transform the eigenvalue problem solving to a series of corresponding boundary value problem solving and eigenvalue ...

In this work, we proposed a multigrid finite element (MFE) method for solving the Fredholm integral eigenvalue problems. The main motivation for such studies is to compute the Karhunen-Loève expansions of random fields, which play an important role in ...