论文标题
基于Arnoldi的正交和分层的无差异多项式基础及其应用
Arnoldi-based orthonormal and hierarchical divergence-free polynomial basis and its applications
论文作者
论文摘要
本文提出了一种方法，用于构建单纯性（2D中的三角形和3D中的三角形）中任意程度的无差异多项式基础的方法。它允许快速计算某些PDE的所有数值解决方案，从零度到指定度\ textit {k}。在有限精确的算术中，产生的无发散基础是正顺序，层次和健壮的。核心是基于阿诺迪的程序。它构建了小于或等于\ textit {k}的多维多项式的矫正和层次基础。通过结合这些多项式基础函数来产生无差异基础。使用这些基础功能开发了杂交BDM混合方法的有效实现。层次结构允许使用仅针对Lem \ textIt {k}计算的本地问题解决方案来增量全局矩阵（零至textit {k}）的全局向量（零至\ textit {k}）。正常性和无差异属性简化了本地问题。考虑的PDE是光滑域和角域中的Helmholtz，Laplace和Poisson问题。这些优点扩展到其他PDE，例如不可压缩的Stokes，不可压缩的Navier-Stokes和Maxwell方程。
This paper presents a methodology to construct a divergence-free polynomial basis of an arbitrary degree in a simplex (triangles in 2D and tetrahedra in 3D) of arbitrary dimension. It allows for fast computation of all numerical solutions from degree zero to a specified degree \textit{k} for certain PDEs. The generated divergence-free basis is orthonormal, hierarchical, and robust in finite-precision arithmetic. At the core is an Arnoldi-based procedure. It constructs an orthonormal and hierarchical basis for multi-dimensional polynomials of degree less than or equal to \textit{k}. The divergence-free basis is generated by combining these polynomial basis functions. An efficient implementation of the hybridized BDM mixed method is developed using these basis functions. Hierarchy allows for incremental construction of the global matrix and the global vector for all degrees (zero to \textit{k}) using the local problem solution computed just for degree \textit{k}. Orthonormality and divergence-free properties simplify the local problem. PDEs considered are Helmholtz, Laplace, and Poisson problems in smooth domains and in a corner domain. These advantages extend to other PDEs such as incompressible Stokes, incompressible Navier-Stokes, and Maxwell equations.
