README.md

k = 0: The Neumann Problem for the Axisymmetric Poisson Equation

Usage

Syntax

Lowest Order Program:

To compare a known exact solution u and to its approximated solution:

[err] = weighted_HL_k_0_p1_e(f,grad_f_r,grad_f_z,gd,sf,ns,mesh,n,u,grad_u_r,grad_u_z)

To find the approximated solution to an unknown solution:

[basis,u_h] = weighted_HL_k_0_p1(f,grad_f_r,grad_f_z,gd,sf,ns,mesh,n)

First Order Program:

To compare a known exact solution u and to its approximated solution:

[err,grad_err,max_err] = weighted_HL_k_0_p2_e(f,grad_f_r,grad_f_z,gd,sf,ns,mesh,n,u,grad_u_r,grad_u_z)

To find the approximated solution to an unknown solution:

[basis,u_h] = weighted_HL_k_0_p2(f,grad_f_r,grad_f_z,gd,sf,ns,mesh,n)

Inputs

f - given function
grad_f_r - gradient(f) with respect to r
grad_f_z - gradient(f) with respect to z
gd,sf,ns - outputs of pdepoly specifying domain
mesh - max mesh level
n - n-th Fourier mode
u - exact solution function
grad_u_r - gradient(u) with respect to r
grad_u_z - gradient(u) with respect to z

Outputs

err - array of L2 errors for mesh levels corresponding to indices
grad_err - array of L2 gradient errors for mesh levels corresponding to indices
max_err - array of max errors for mesh levels corresponding to indicies
basis - a matrix representing piece-wise basis functions for each node in each triangle. basis(i,:,k) represents the pieceiwise basis function for the ith node in triangle k.
u_h - approximated solution vector for u

Example

% add path for get_data_7() function
addpath ../data/
% define the highest mesh level
mesh = 5;
% define the nth-Fourier mode
n = 1;
% define the problem domain
pdepoly([0,1,1,0], [0,0,1,1]);
% define the equations
[f,grad_f_r,grad_f_z,u,grad_u_r,grad_u_z] = get_data7();
% run the program
% the lowest order
[err] = weighted_HL_k_0_p1_e(f,grad_f_r,grad_f_z,gd,sf,ns,mesh,n,u,grad_u_r,grad_u_z);
% or the first order
[err,grad_err,max_err] = weighted_HL_k_0_p2_e(f,grad_f_r,grad_f_z,gd,sf,ns,mesh,n,u,grad_u_r,grad_u_z);

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