Mechanical quadrature method and splitting extrapolation algorithm for the boundary integral equation of the axisymmetric anisotropic Darcy's equation

2015 ◽  
Vol 94 (1) ◽  
pp. 135-150 ◽  
Author(s):  
Hu Li ◽  
Jin Huang
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equation ◽  
Boundary Integral ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Extrapolation Algorithm ◽  
Splitting Extrapolation ◽  
Mechanical Quadrature Method

scholarly journals Mechanical Quadrature Method and Splitting Extrapolation for Solving Dirichlet Boundary Integral Equation of Helmholtz Equation on Polygons

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Hu Li ◽  
Yanying Ma
Keyword(s):  
Integral Equation ◽  
Helmholtz Equation ◽  
Boundary Integral Equation ◽  
Dirichlet Boundary ◽  
Boundary Integral ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Splitting Extrapolation ◽  
Accuracy Order ◽  
Mechanical Quadrature Method

We study the numerical solution of Helmholtz equation with Dirichlet boundary condition. Based on the potential theory, the problem can be converted into a boundary integral equation. We propose the mechanical quadrature method (MQM) using specific quadrature rule to deal with weakly singular integrals. Denote byhmthe mesh width of a curved edgeΓm  (m=1,…,d)of polygons. Then, the multivariate asymptotic error expansion of MQM accompanied withO(hm3)for all mesh widthshmis obtained. Hence, once discrete equations with coarse meshes are solved in parallel, the higher accuracy order of numerical approximations can be at leastO(hmax⁡5)by splitting extrapolation algorithm (SEA). A numerical example is provided to support our theoretical analysis.

scholarly journals A Mechanical Quadrature Method for Solving Delay Volterra Integral Equation with Weakly Singular Kernels

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Li Zhang ◽  
Jin Huang ◽  
Yubin Pan ◽  
Xiaoxia Wen
Keyword(s):  
Integral Equation ◽  
Volterra Integral Equation ◽  
Richardson Extrapolation ◽  
Quadrature Method ◽  
Mechanical Quadrature ◽  
Proportional Delays ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Singular Kernels ◽  
Mechanical Quadrature Method

In this work, a mechanical quadrature method based on modified trapezoid formula is used for solving weakly singular Volterra integral equation with proportional delays. An improved Gronwall inequality is testified and adopted to prove the existence and uniqueness of the solution of the original equation. Then, we study the convergence and the error estimation of the mechanical quadrature method. Moreover, Richardson extrapolation based on the asymptotic expansion of error not only possesses a high accuracy but also has the posterior error estimate which can be used to design self-adaptive algorithm. Numerical experiments demonstrate the efficiency and applicability of the proposed method.

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