Abstract ：

This study used a multigrid-based convolutional neural network architecture known as MgNet in operator learning to solve numerical partial differential equations (PDEs). Given the property of smoothing iterations in multigrid methods where low-frequency errors decay slowly, we introduced a low-frequency correction structure for residuals to enhance the standard V-cycle MgNet. The enhanced MgNet model can capture the low-frequency features of solutions considerably better than the standard V-cycle MgNet. The numerical results obtained using some standard operator learning tasks are better than those obtained using many state-of-the-art methods, demonstrating the efficiency of our model. Moreover, numerically, our new model is more robust in case of low- and high-resolution data during training and testing, respectively.

Keyword ：

Low-frequency correction Low-frequency correction Numerical partial differential equations Numerical partial differential equations Operator learning Operator learning MgNet MgNet

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Abstract ：

In this paper, a discontinuous Galerkin (DG) time stepping method combined with the standard finite element method in space is proposed to solve a class of semilinear parabolic differential equations with time constant delay. The time semi-discretization and the relevant global convergence of the DG solution under suitable uniform meshes are derived. The standard Galerkin method in space is used to obtain the fully discrete scheme and the optimal global convergence of the full discretization is presented. Numerical experiments for one-dimensional and two-dimensional equations are provided to demonstrate the theoretical results.

Keyword ：

Finite element method Finite element method Time constant delay Time constant delay Discontinuous Galerkin time stepping Discontinuous Galerkin time stepping Fully discrete scheme Fully discrete scheme Parabolic equations Parabolic equations

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Abstract ：

This paper aims to introduce a novel adaptive multigrid method for the elasticity eigenvalue problem. Different from the developing adaptive algorithms for the elasticity eigenvalue problem, the proposed approach transforms the elasticity eigenvalue problem into a series of boundary value problems in the adaptive spaces and some small-scale elasticity eigenvalue problems in a low-dimensional space. As our algorithm avoids solving large-scale elasticity eigenvalue problems, which is time-consuming, and the boundary value problem can be solved efficiently by the adaptive multigrid method, our algorithm can evidently improve the overall solving efficiency for the elasticity eigenvalue problem. Meanwhile, we present a rigorous theoretical analysis of the convergence and optimal complexity. Finally, some numerical experiments are presented to validate the theoretical conclusions and verify the numerical efficiency of our approach.

Keyword ：

Elasticity eigenvalue problem Elasticity eigenvalue problem Adaptive multigrid method Adaptive multigrid method Convergence and optimal complexity Convergence and optimal complexity Finite element method Finite element method

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Abstract ：

In this paper, we propose a type of adaptive multigrid method for eigenvalue problem based on the multilevel correction method and adaptive multigrid method. Different from the standard adaptive finite element method applied to eigenvalue problem, with our method we only need to solve a linear boundary value problem on each adaptive space and then correct the approximate solution by solving a low dimensional eigenvalue problem. Further, the involved boundary value problems are solved by some adaptive multigrid iteration steps. The proposed adaptive algorithm can reach the same accuracy as the standard adaptive finite element method for eigenvalue problem but evidently reduces the computational work. In addition, the corresponding convergence and optimal complexity analysis are derived theoretically and numerically, respectively.

Keyword ：

convergence convergence optimal complexity optimal complexity Eigenvalue problem Eigenvalue problem adaptive multigrid method adaptive multigrid method multilevel correction multilevel correction

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In this paper, we study an efficient multigrid method to solve the semilinear interface problems. We first give an optimal finite element error estimate for the semilinear interface problems under a weak condition for the nonlinear term compared with the existing conclusions. Then next based on the finite element error estimate, we design a novel multigrid method for semilinear elliptic problems. The proposed multigrid method only requires to solve a linear interface problem in each level of the multilevel space sequence and a small-scale semilinear interface problem in a correction space. The involved linear interface problem can be solved efficiently by the multigrid iteration. The dimension of the correction space is small and fixed, which is independent from the fine spaces. Thus the computational time of the correction step is negligible compared with that of the linear interface problems in the fine spaces. On the whole, the efficiency of the presented multigrid method is nearly the same as that of the multigrid method for linear interface problems. Additionally, unlike the existing finite element error estimates and the multigrid methods for semilinear interface problems, which always require the bounded second order derivatives of the nonlinear terms, all the analysis in our paper only requires a Lipschitz continuous condition. (C) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

Keyword ：

Multigrid method Multigrid method Finite element method Finite element method Semilinear interface problem Semilinear interface problem

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Abstract ：

In this study, a multilevel correction-type goal-oriented adaptive finite element method is designed for semilinear elliptic equations. Concurrently, the corresponding convergence property is theoretically proved. In the novel goal-oriented adaptive finite element method, only a linearized primal equation and a linearized dual equation are required to be solved in each adaptive finite element space. To ensure convergence, the approximate solution of the primal equation was corrected by solving a small-scale semilinear elliptic equation after the central solving process in each adaptive finite element space. Since solving of the large-scale semilinear elliptic equations is avoided and the goal-oriented technique is absorbed, there has been a significant improvement in the solving efficiency for the goal functional of semilinear elliptic equations. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

Keyword ：

Adaptive finite element method Adaptive finite element method Multilevel correction method Multilevel correction method Convergence Convergence Goal-oriented Goal-oriented

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In this study, a multilevel local defect-correction method is designed for solving nonsymmetric eigenvalue problems. The main feature of our approach is the transformation of the nonsymmetric eigenvalue problems into several symmetric boundary value problems defined in a multilevel finite element space sequence and some low-dimensional nonsymmetric eigenvalue problems defined in a specially designed correction space. Moreover, the symmetric boundary value problems involved in our algorithm are solved by the local defect-correction strategy that divides the computing domain into small-scale subdomains. Since solving the high-dimensional nonsymmetric eigenvalue problems is avoided which is quite time-consuming compared with that of solving boundary value problems, the presented algorithm greatly improves the solving efficiency for nonsymmetric eigenvalue problems. Rigorous theoretical analysis and several numerical experiments are given to demonstrate the efficiency of our algorithm.

Keyword ：

Multilevel correction method Multilevel correction method Nonsymmetric eigenvalue problems Nonsymmetric eigenvalue problems Local defect-correction method Local defect-correction method

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In this paper, we study numerical methods for solving a multi-dimensional fracture model, which couples n-dimensional Darcy flow in matrix with (n - 1)-dimensional Brinkman flow on fracture. A two-grid decoupled algorithm is proposed, in which the mixed model is decoupled by using the coarse grid approximation to the interface conditions, and then efficient singlemodel solvers are applied for decoupled Darcy and Brinkman problems on the fine mesh. Error estimates show that the two-grid decoupled algorithm retains the same order of approximation accuracy as the coupled one. Numerical experiments in two-dimensional (2D) and three-dimensional (3D) geometries are conducted, and their results confirm our theoretical analysis to illustrate the efficiency and effectiveness of the proposed method for solving multi-domain problems.

Keyword ：

Two-grid decoupled algorithm Two-grid decoupled algorithm Error estimates Error estimates Darcy-Brinkman model Darcy-Brinkman model Numerical experiments Numerical experiments

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Abstract ：

The Peng-Robison equation of state, one of the most extensively applied equations of state in the petroleum industry and chemical engineering, has an excellent appearance in predicting the thermodynamic properties of a wide variety of materials. It has been a great challenge on how to design numerical schemes with preservation of mass conservation and energy dissipation law. Based on the exponential time difference combined with the stabilizing technique and added Lagrange multiplier enforcing the mass conservation, we develop the efficient first-and second-order numerical schemes with preservation of maximum bound principle (MBP) to solve the single-component two-phase diffuse interface model with Peng-Robison equation of state. Convergence analyses as well as energy stability are also proven. Several twodimensional and three-dimensional experiments are performed to verify these theoretical results.

Keyword ：

maximum bound principle maximum bound principle exponential time differencing exponential time differencing Peng-Robinson equation of state Peng-Robinson equation of state diffuse interface model diffuse interface model Lagrange multiplier Lagrange multiplier

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In this paper, we discuss the superconvergence of the "interpolated" collocation solutions for weakly singular Volterra integral equations of the second kind. Based on the collocation solution u(h), two different interpolation postprocessing approximations of higher accuracy: I-2h(2m-1) u(h) based on the collocation points and I(2h)(m)u(h) based on the least square scheme are constructed, whose convergence order are the same as that of the iterated collocation solution. Such interpolation postprocessing methods are much simpler in computation. We further apply this interpolation postprocessing technique to hybrid collocation solutions and similar results are obtained. Numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing methods.

Keyword ：

Interpolation postprocessing Interpolation postprocessing Weakly singular kernels Weakly singular kernels Volterra integral equations Volterra integral equations Supercloseness Supercloseness Superconvergence Superconvergence Collocation Collocation Hybrid collocation Hybrid collocation

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