An adaptive finite element DtN method for the open cavity scattering problems

弹性波散射问题中自适应有限元DtN方法的收敛性

Convergence of an adaptive finite element DtN method for the elastic wave scattering problem

本文研究了弹性波散射问题中基于DtN算子的自适应有限元方法的收敛性，通过发展新的对偶论证，推导了有限元方法中截断DtN算子的后验误差估计，该估计包含有限元近似误差和截断误差，后者随截断参数指数衰减。提出了一种自适应有限元算法来求解弹性障碍散射问题，并通过数值实验验证了该方法的有效性。

周期结构中声-弹相互作用问题的自适应有限元DtN方法

An Adaptive Finite Element DtN Method for the Acoustic-Elastic Interaction Problem in Periodic Structures

针对周期结构中声-弹相互作用问题，提出一种基于后验误差估计的自适应有限元DtN方法，该方法结合声波和弹性波DtN算子，并利用指数衰减的截断误差指导网格自适应，数值实验验证了其有效性。

针对双周期结构中麦克斯韦方程的自适应有限元DtN方法

An Adaptive Finite Element DtN Method for Maxwell's Equations in Biperiodic Structures

本文提出一种自适应有限元DtN方法，用于解决双周期结构中电磁波的衍射问题，通过后验误差估计控制计算误差，并通过数值实验验证了该方法的有效性。

用于亥姆霍兹方程的方向自适应平面波间断伽略金方法

Direction-Adaptive Plane-Wave Discontinuous Galerkin Methods for the Helmholtz Equation

该研究提出方向自适应平面波间断伽略金方法，通过最小化残差选择传播方向，经数值实验验证其可准确恢复亥姆霍兹方程解的主导相位方向，对低方向复杂度解效果最优。

An adaptive finite element PML method for the open cavity scattering problems

Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper is concerned with the numerical solutions of the transverse electric and magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite element method with the pe…

POD model order reduction with space-adapted snapshots for incompressible flows

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the ve…

无标签数据下用于代理建模的深度自适应采样

Deep adaptive sampling for surrogate modeling without labeled data

本文提出深度自适应采样方法DAS²，将残差损失视为未归一化概率密度并用深度生成模型采样以细化训练集，从而为低正则性参数微分方程构建无标签数据的代理模型，并通过参数化二维空腔流动等实验验证其有效性。

Analysis of finite element approximations of Stokes equations with non-smooth data

In this paper we analyze the finite element approximation of the Stokes equations with non-smooth Dirichlet boundary data. To define the discrete solution, we first approximate the boundary datum by a smooth one and then apply a standard finite element method to the regularized problem. We prove almost optimal order error estimates for two regularization procedures in the case of general data in fractional order Sobolev spaces, and for the Lagrange interpolation (with appropriate modifications at the discontinuities) for piecewise smooth data. Our results apply in particular to the classic lid…

A Geometric Multigrid Preconditioning Strategy for DPG System Matrices

The discontinuous Petrov-Galerkin (DPG) methodology of Demkowicz and Gopalakrishnan [15,17] guarantees the optimality of the solution in an energy norm, and provides several features facilitating adaptive schemes. A key question that has not yet been answered in general - though there are some results for Poisson, e.g. - is how best to precondition the DPG system matrix, so that iterative solvers may be used to allow solution of large-scale problems. In this paper, we detail a strategy for preconditioning the DPG system matrix using geometric multigrid which we have implemented as part of Came…

用于模拟微波加热流动的不连续Petrov-Galerkin有限元框架

A discontinuous Petrov-Galerkin finite-element framework for the simulation of microwave-heated flows

该研究提出耦合DPG有限元方法与连续Galerkin有限元方法的高阶多物理场求解器，经基准验证可用于微波加热流动模拟，内存低、可扩展，适用于大规模并行模拟与微波辅助化学过程优化。

用于带 Dirichlet 边界条件的不可压缩 Navier-Stokes 方程的四阶及更高阶有限元方法

Fourth- and higher-order finite element methods for the incompressible Navier-Stokes equations with Dirichlet boundary conditions

本文基于 GePUP 形式构建线方法高阶有限元求解器，结合 IMEX Runge-Kutta、几何多重网格与自适应网格，实现带 Dirichlet 条件 INSE 的高阶时空精确模拟。