论文标题
$ q^3 $ suppral元素方法的单调性用于离散laplacian
Monotonicity of $Q^3$ spectral element method for discrete Laplacian
论文作者
论文摘要
离散laplacian的单调性，即刚度矩阵的逆阳性，意味着离散的最大原理，这通常是在非结构化网格上的高阶准确方案而言是不正确的。另一方面，可以在结构化网格上构造高阶精确单调方案。所有先前已知的高阶精度逆正方案都是第四阶准确方案，该方案是M-Matrix或两个M-矩阵的产物。对于二维Laplacian的$ Q^3 $频谱元素方法，我们证明其刚度矩阵是四个M矩阵的产物，因此它是单调的。这样的方案可以被视为第五阶准确的有限差异方案。
The monotonicity of discrete Laplacian, i.e., inverse positivity of stiffness matrix, implies discrete maximum principle, which is in general not true for high order accurate schemes on unstructured meshes. On the other hand, it is possible to construct high order accurate monotone schemes on structured meshes. All previously known high order accurate inverse positive schemes are fourth order accurate schemes, which is either an M-matrix or a product of two M-matrices. For the $Q^3$ spectral element method for the two-dimensional Laplacian, we prove its stiffness matrix is a product of four M-matrices thus it is monotone. Such a scheme can be regarded as a fifth order accurate finite difference scheme.
