A High-Accuracy Mechanical Quadrature Method for Solving the Axisymmetric Poisson's Equation (Advances in Applied Mathematics and Mechanics, 2018)
In this article, we consider the numerical solution for Poisson's equation inaxisymmetric geometry. When the boundary condition and source term are axisymmetric,the problem reduces to solving Poisson's equation in cylindrical coordinates inthe two-dimensional ($r,z$) region of the original three-dimensional domain $S$. Hence,the original boundary value problem is reduced to a two-dimensional one. To makeuse of the Mechanical quadrature method (MQM), it is necessary to calculate a particularsolution, which can be subtracted off, so that MQM can be used to solve the resultingLaplace problem, which possesses high accuracy order $\mathcal{O}$($h^3_{max}$) and low computingcomplexities. Moreover, the multivariate asymptotic error expansion of MQMaccompanied with $\mathcal{O}$($h^3_i$) for all mesh widths $h_i$ is got. Hence, once discrete equationswith coarse meshes are solved in parallel, the higher accuracy order of numerical approximationscan be at least $\mathcal{O}$($h^5_{max}$) by the splitting extrapolation algorithm (SEA).Meanwhile, a posteriori asymptotic error estimate is derived, which can be used toconstruct self-adaptive algorithms. The numerical examples support our theoreticalanalysis.