Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials
Efficient Spectral-Galerkin Method I. Direct Solvers of Second- and Fourth-Order Equations Using Legendre Polynomials
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DOI：
10.1137/0915089
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发表时间：
1994-11
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影响因子：
--
通讯作者：
Jie Shen
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文献类型：
--
作者：
Jie Shen
本文提出了基于legende - galerkin近似的二阶和四阶椭圆方程直接解的几种有效算法。这些算法效率的关键是构造合适的基函数，从而导致离散变分公式具有稀疏矩阵的系统。对于具有$(N - 1)^d $未知数的d维域，算法的复杂性是$N^{d + 1} $运算的一个小倍数，而对于具有光滑解的问题，算法的收敛速度是指数级的。此外，由于算法的瓶颈是矩阵-矩阵乘法，因此算法可以有效地并行化。
This paper presents some efficient algorithms based on the Legendre–Galerkin approximations for the direct solution of the second- and fourth-order elliptic equations. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with sparse matrices for the discrete variational formulations. The complexities of the algorithms are a small multiple of $N^{d + 1} $ operations for a d-dimensional domain with $(N - 1)^d $ unknowns, while the convergence rates of the algorithms are exponential for problems with smooth solutions. In addition, the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix-matrix multiplications.