论文标题
雅各比光谱搭配方法的弱奇异非局部扩散方程的融合分析具有体积约束
Convergence analysis of Jacobi spectral collocation methods for weakly singular nonlocal diffusion equations with volume constraints
论文作者
论文摘要
本文考虑了具有Dirichlet型体积限制的弱奇异非局部扩散方程的有效光谱溶液。我们认为的方程式包含一个积分运算符，该集成运算符通常在积分域的中点具有奇异性，并且积分运算符的近似值是求解非局部方程的必要困难之一。为了克服这个问题，提出了双面的雅各比光谱正交规则，以开发一种用于非局部扩散方程的雅各比光谱搭配方法。提出了对所提出的方法对$ l^\ infty $ norm的严格合并分析，我们进一步证明了雅各比搭配解决方案将其相应的局部限制收敛，因为非本地相互作用消失了。给出了数值示例以验证理论结果。
This paper considers efficient spectral solutions for weakly singular nonlocal diffusion equations with Dirichlet-type volume constraints. The equation we consider contains an integral operator that typically has a singularity at the midpoint of the integral domain, and the approximation of the integral operator is one of the essential difficulties in solving nonlocal equations. To overcome this problem, two-sided Jacobi spectral quadrature rules are proposed to develop a Jacobi spectral collocation method for nonlocal diffusion equations. A rigorous convergence analysis of the proposed method with the $L^\infty$ norm is presented, and we further prove that the Jacobi collocation solution converges to its corresponding local limit as nonlocal interactions vanish. Numerical examples are given to verify the theoretical results.
