Lars Diening, Johannes Storn
This paper introduces an ultra-weak space-time DPG method for the heat equation. We prove well-posedness of the variational formulation with broken test functions and verify quasi-optimality of a practical DPG scheme. Numerical experiments visualize beneficial properties of an adaptive and parabolically scaled mesh-refinement driven by the built-in error control of the DPG method.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A time-stepping DPG scheme for the heat equation
本文提出并分析了热方程的 discontinuous Petrov-Galerkin 方法，基于后向欧拉时间步长和超弱变分公式，证明了方法的稳定性并推导了 Céa 型误差估计。
In this work, we propose a new quasi-optimal test norm for a discontinuous Petrov-Galerkin (DPG) discretization of the ultra-weak formulation of the convection-diffusion equation. We prove theoretically that the proposed test norm leads to bounds between the target norm and the energy norm induced by the test norm which are robust with respect to the diffusion parameter in the solution and gradient components and have favorable scalings in the trace components. We conclude with numerical experiments to confirm our theoretical results.
We analyse backward Euler time stepping schemes for the primal DPG formulation of a class of parabolic problems. Optimal error estimates are shown in the natural norm and in the $L^2$ norm of the field variable. For the heat equation the solution of our primal DPG formulation equals the solution of a standard Galerkin scheme and, thus, optimal error bounds are found in the literature. In the presence of advection and reaction terms, however, the latter identity is not valid anymore and the analysis of optimal error bounds requires to resort to elliptic projection operators. It is essential tha…
DPG method with optimal test functions for a fractional advection diffusion equation
本文提出了一种分数对流扩散问题的超弱变分公式，并证明其适定性，基于该公式定义了具有最优测试函数的DPG近似，并展示了其近优收敛性，数值实验验证了均匀和自适应细化网格下的预期收敛特性。
A spacetime DPG method for the wave equation in multiple dimensions
本文提出了一种用于多维波动方程的时空断续Petrov-Galerkin（DPG）方法，基于使用断裂图空间的弱形式，并通过已有的抽象框架证明了该形式的正确性。通过证明密度结果，详细讨论了简单域在任意维度中的情况。然后研究了基于该弱形式的DPG方法的理论和数值特性，给出了三角形、矩形、四面体和六面体时空域网格的误差估计和数值结果，并探讨了DPG方法内置误差估计器在二维和三维中的自适应网格细化策略的潜力。
Dispersive and dissipative errors in the DPG method with scaled norms for Helmholtz equation
研究DPG方法在亥姆霍兹方程中的应用，发现缩放参数接近极限值时性能更优，通过分散分析揭示了方法中复离散波数导致的数值耗散，对比了非标准最小二乘 Galerkin 方法与标准方法的性能。
A spacetime DPG method for the Schrodinger equation
本文提出了一种用于线性时间依赖薛定谔方程的空间时间不连续Petrov-Galerkin (DPG)方法，通过使用二阶薛定谔算子来捕捉不规则解，并分析了基于弱形式的DPG方法的收敛性。
Breaking spaces and forms for the DPG method and applications including Maxwell equations
本文研究了DPG方法中打破空间的应用，针对麦克斯韦方程提出完整误差分析，并简化了其他方程的分析，证明了不同形式的稳定性等价性。
Ultra-weak formulation of a hypersingular integral equation on polygons and DPG method with optimal test functions
本文提出多边形上超奇异积分方程的多重弱形式，证明其正确性与标准变分形式等价，并基于此提出最优测试函数的断续Petrov-Galerkin方法，证明其在L²空间的近似最优收敛性，通过数值实验验证理论结果。
Multistage DPG time-marching scheme for nonlinear problems
本文针对非线性常微分方程和抽象发展方程，提出基于DPG格式的多级时间推进方法，通过级间后处理避免额外指数运算，比经典指数Rosenbrock方法更高效，并证明方法可达二至四阶精度。
Convergence rates of the DPG method with reduced test space degree
本文基于Aubin-Nitsche对偶定理探讨了DPG方法在弱范数下的更高收敛率，揭示了降阶测试空间下DPG方法的可解性及非一致性分析方法。
The DPG methodology applied to different variational formulations of linear elasticity
本文通过不同变分公式求解线性弹性方程，展示DPG方法的灵活性，证明传统能量空间下部分公式不相容，但使用断裂能量空间和界面变量后仍保持良好设定。
An analysis of the practical DPG method
本文对实际实现中的不连续Petrov-Galerkin（DPG）方法进行了完整的误差分析，证明了当试验空间采用p次多项式时，使用r≥p+N（N为空间维数）的多项式近似测试空间可保持最优收敛率，并讨论了刚度矩阵的条件数。
In this work, we propose and develop an arbitrary-order adaptive discontinuous Petrov-Galerkin (DPG) method for the nonlinear Grad-Shafranov equation. An ultraweak formulation of the DPG scheme for the equation is given based on a minimal residual method. The DPG scheme has the advantage of providing more accurate gradients compared to conventional finite element methods, which is desired for numerical solutions to the Grad-Shafranov equation. The numerical scheme is augmented with an adaptive mesh refinement approach, and a criterion based on the residual norm in the minimal residual method i…
On the stability of DPG formulations of transport equations
本文研究了传输方程的不连续Petrov-Galerkin形式的稳定性，证明了在一定条件下，使用高阶多项式试函数空间可得到均匀稳定的离散化方案。
Fortin operators for DPG advection discretizations
该研究针对任意空间维度和多项式次数的平流方程DPG离散化构造了Fortin算子，证明其相关性质并应用于全离散残差最小化方法，得到拟最优性等结论。
Formulation and analysis of a DPG discretization for a simplified electrochemical model
针对简化电化学电池模型，研究人员构建并分析了DPG离散化方案，证明了相关变分公式适定性与方法收敛性，为电池高效分析设计提供了新的数值工具。
We consider the finite element (FE) approximation of the shallow water equations (SWE) by considering discretizations in which both space and time are established using an unconditionally stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to break the test space and achieve unconditionally stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow…
A Vectorial Envelope Maxwell Formulation for Electromagnetic Waveguides with Application to Nonlinear Fiber Optics
本文提出一种超弱 DPG 公式求解多模光纤波导中电磁场矢量包络的时谐 Maxwell 方程组，通过引入包络假设降低纵向振荡性，使可计算的光纤长度提升 1000 倍，并成功应用于掺镱放大器的热致横模不稳定性模拟。
In its application to the modeling of a mineral separation process, we propose the numerical analysis of the Cahn-Hilliard equation by employing spacetime discretizations of the automatic variationally stable finite element (AVS-FE) method. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov-Galerkin (DPG) method by Demkowicz and Gopalakrishnan. The trial space, however, consists of globally continuous Hilbert spaces such as H1 and H(div). Hence, the AVS-FE approximations employ classical C0 or Raviart-Thom…
The Numerical Properties of G-heat equation and Related Application
本文研究G-热方程的数值收敛性，证明牛顿迭代收敛和全隐式离散的单调性与稳定性，并得出其收敛到粘性解的结论。
Splitting schemes for hyperbolic heat conduction equation
针对双曲热传导方程，构造了方向分裂的加性格式，证明了局部一维分裂格式的无条件稳定性，并提出了基于温度和热流方程组的新的分裂格式。
We consider the Virtual Element discretization of the Navier-Stokes equations coupled with the heat equation where the viscosity depends on the temperature. We present the Virtual Element discretization of the coupled problem, show its well-posedness, and prove optimal error estimates. Numerical experiments which confirm the theoretical error bounds are also presented.
A generic Scheme For the time-dependent Navier-Stokes Equation Coupled With The Heat Equation
本文针对黏度随温度变化的时变Navier-Stokes方程与热方程耦合问题，提出一种梯度离散化方法，在无非物理条件下证明其收敛性，并通过数值实验验证了理论结果。
Two unconditional stable schemes for simulation of heat equation on manifold using DEC
本文利用离散微分几何技术，提出了两种用于流形上热方程模拟的无条件稳定数值格式，并通过最大原理分析了其稳定性和误差。
Analytical and numerical treatment of the heat conduction equation obtained via time-fractional distributed-order heat conduction law
本文通过能量平衡方程与分数阶本构热传导定律构建热传导方程，利用傅里叶和拉普拉斯变换及有限差分法分析求解，并通过数值例子验证了两种方法在多项式分布阶热传导定律下的适用性与一致性。
Constructive a priori error estimates for a full discrete approximation of periodic solutions for the heat equation
本文研究了热方程在时间周期条件下全离散数值解的构造性a priori误差估计，重点在于误差分析和理论保证。
We introduce an unconditionally stable finite element (FE) method, the automatic variationally stable FE (AVS-FE) method for the numerical analysis of the Korteweg-de Vries (KdV) equation. The AVS-FE method is a Petrov-Galerkin method which employs the concept of optimal discontinuous test functions of the discontinuous Petrov- Galerkin (DPG) method. However, since AVS-FE method is a minimum residual method, we establish a global saddle point system instead of computing optimal test functions element-by-element. This system allows us to seek both the approximate solution of the KdV initial bou…
We introduce a time-implicit, finite-element based space-time discretization scheme for the backward stochastic heat equation, and for the forward-backward stochastic heat equation from stochastic optimal control, and prove strong rates of convergence. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。