Computational Science Asked on October 23, 2021
I am trying to solve the Poisson equation numerically using the FDM method in C++. But I have a little confusion with the iterative process. I understand that the number of iterations should go until the solution converges, but how to calculate if the error is greater/less than the tolerance level? Here is a small piece of code I tried in C++ but it is flawed. I checked some other codes posted online and some has calculated the average of the residual values in a matrix and checked accordingly. I’d appreciate it if anyone could help me with the concept.
void calculate_voltage(){
voltage_initialization(); //creating a matrix V and initilizing with Dirchilet's boundary condition
double tolerance = pow(10,-5);
bool done = true;
int itr = 0;
double pi = 3.14;
double t = cos(pi/nx) + cos(pi/ny);
double omega = (8 - sqrt(64 - 16*pow(t,2)))/(pow(t,2)); //relaxation parameter
while(done == true){
itr ++;
for(int i = 1;i<nx-1;i++){
for(int j = 1;j<ny-1;j++)
{
double vv = (V[i-1][j] + V[i+1][j] + V[i][j-1] + V[i][j+1] + step_size_ * source[i][j])/4.0;
double R = vv - V[i][j]; //residual for SOR
if(abs(R) <= tolerance){done = false;} //to check if the correction converges or not
V[i][j] = V[i][j] + omega* R; //new V
}
}
}
}
Since you're solving the linear poisson equation $Ax = b$ I'd just check that the L2-norm of the residual vector illustrates convergence. I think the best thing to do is to calculate the initial norm $rho_0 = ||b||_2$ and then have two tolerances, one relative ($epsilon_r$) and one absolute ($epsilon_a$) and you would terminate if either of them is satisfied. So if $frac{rho_i}{rho_0} < epsilon_r$ or if $rho_i < epsilon_a$ where $rho_i = ||b - Ax_i||_2$. You could also use the max value of the residual to check (the infinity norm), but the most commonly used ones is the 2-norm, and this extends nicely to GMRES and other krylov solvers. regarding actual values, I'd say try an initial absolute tolerance of $1e-14$ and relative of $1e-12$ and see if that works. You can also plot convergence as a function of iterations which may be instructive.
Answered by EMP on October 23, 2021
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