Abstract
The stationary fractional advection dispersion equation is discretized by linear finite element scheme, and a full V-cycle multigrid method (FV-MGM) is proposed to solve the resulting system. Some useful properties of the approximation and smoothing operators are proved. Using these properties we derive the convergence results in both <svg style="vertical-align:-0.0pt;width:16.1625px;" id="M1" height="15.8375" version="1.1" viewBox="0 0 16.1625 15.8375" width="16.1625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g> <g transform="matrix(.012,-0,0,-.012,9.763,7.613)"><path id="x32" d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" /></g> </svg> norm and energy norm for FV-MGM. Numerical examples are given to demonstrate the convergence rate and efficiency of the method.
Highlights
We investigate the finite element full V-cycle multigrid method (FV-MGM) to the boundary value problem of linear stationary fractional advection dispersion equation (FADE)
By selecting appropriate iteration operator and iteration numbers, we prove that FV-MGM has the same convergence rate as classic FEM and the computational cost increases linearly with respect to the increasing of unknown variables
Theorems and examples in this paper show that the convergence rate of FVMGM is the same as classic FEM under the stopping criterion (53), and the computational work is only O(dim Vk) while the stopping criterion is taken (57)
Summary
We investigate the finite element full V-cycle multigrid method (FV-MGM) to the boundary value problem of linear stationary fractional advection dispersion equation (FADE). Many scholars developed numerical methods, including finite difference method [11], finite element method [12,13,14], spectral method [15] and moving collocation method [16] to solve FADEs. Most of them used Gauss elimination method or conjugate gradient norm residual method to solve the resulting system, so the computational complexity is O(N3) or O(N log2N). We follow the ideas in [17, 18] to develop a FV-MGM for solving the resulting system of Problem 1 discretized by linear finite element method.
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