RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels

De Marchi S.; Martinez A.; Perracchione E.; Rossini M.

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RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels

机译：椭圆PDE的基于RBF的统一方法划分：通过可变尺度的核的适应性和稳定性问题

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We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure.

机译：我们研究基于径向基函数的单位搭配分区对泊松方程逼近的适应性问题。自适应残差二次采样方法是使用准均匀节点序列执行的，从而导致了一种灵活的工具，但是由于配置矩阵的条件不佳，该工具可能会出现数值不稳定的情况。因此，我们开发了一种混合方法，该方法利用了所谓的可变比例内核。所提出的算法在数值上确保了自适应过程的收敛性。

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来源
《Journal of Scientific Computing》 |2019年第1期|321-344|共24页
作者
De Marchi S.; Martinez A.; Perracchione E.; Rossini M.;
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作者单位

Univ Padua, Dipartimento Matemat Tullio Levi Civita, Padua, Italy;

Univ Padua, Dipartimento Matemat Tullio Levi Civita, Padua, Italy;

Univ Padua, Dipartimento Matemat Tullio Levi Civita, Padua, Italy;

Univ Milano Bicocca, Dipartimento Matemat & Applicaz, Milan, Italy;

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正文语种 eng
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关键词
Partition of unity method; Radial basis functions; Meshfree approximation; Elliptic PDEs; Variably scaled kernels;

机译：单位方法的划分;径向基函数;无网格近似;椭圆PDE;可变比例的核;

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专利

1. A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs [J] . Griebel M., Schweitzer MA. SIAM Journal on Scientific Computing . 2000,第3期

机译：求解椭圆形，抛物线和双曲型PDE的统一方法的粒子划分
2. Positive constrained approximation via RBF-based partition of unity method [J] . De Rossi Alessandra, Perracchione Emma Journal of Computational and Applied Mathematics . 2017,第期

机译：基于RBF的Unity方法分区的正约束近似
3. Erratum: Structural stability for variable exponent elliptic problems. I. The p(x)-laplacian kind problems and Structural stability for variable exponent elliptic problems. II. The (u)-laplacian and coupled problems (Nonlinear Analysis, Theory, Methods and Applications 72:12 (4649-4660)) [J] . Andreianov E.B., Bendahmane M., Ouaro S. Nonlinear Analysis: An International Multidisciplinary Journal . 2010,第1期

机译：勘误：可变指数椭圆问题的结构稳定性。 I. p（x）-laplacian类问题和可变指数椭圆问题的结构稳定性。二。（u）-拉普拉斯和耦合问题（非线性分析，理论，方法和应用72:12（4649-4660））
4. On h-Adaptivity Analysis Using Multiple Scale Reproducing Kernel Particle Method and Partition of Unity Quadrature [C] . Zhiqian Zhang, Jinxiong Zhou, Qiong Li, Computational Mechanics . 2004

机译：基于多尺度再生核粒子方法的h适应性分析和单位正交划分
5. A Scalable Software Framework for Solving PDES on Distributed Octree Meshes Using Nite Element MethodsA scalable software framework for solving pdes on distributed octree meshes using finite element methods [D] . Lofquist, Alec Dale. 2018

机译：使用NITE元素MetableA可扩展软件框架在分布式Octree网格上求解PDE的可扩展软件框架，用于使用有限元方法在分布式Octree网格上求解PDES
6. A Kernel-free Boundary Integral Method for Elliptic Boundary Value Problems [O] . Wenjun Ying, Craig S. Henriquez -1

机译：椭圆边值问题的无核边界积分方法
7. A Particle-Partition Of Unity Method For The Solution Of Elliptic, Parabolic And Hyperbolic PDEs [O] . Michael Griebel, Marc Alexander Schweitzer 2007

机译：椭圆型，抛物型和双曲型偏微分方程解的统一方法的粒子划分

RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels

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