sharp.c

We search an adapted mesh containing N_{obj} elements which minimizes the total error (error between the numerical solution of the Basilisk Poisson-Helmholtz solver) and the (known) analytical solution.

The solution is an boundary layer-like function.

#include "poisson.h" // solver

#include "./../no_hmin/boundarylayer.h"  // ''problem''
#include "./../../AMR_tools/amr.h"  // AMR
#include "./../../utils/gauss_quadrature.h"  // compute total error

#include "utils.h"

double TOL = 1.e-7;

const face vector alp[] = {D,D};
const scalar lam[] =s;

scalar psi[]; // the numerical solution
psi[left] = dirichlet(exact(x,y,0.));
psi[right] = dirichlet(exact(x,y,0.));
psi[top] = dirichlet(exact(x,y,0.));
psi[bottom] = dirichlet(exact(x,y,0.));

scalar psi_exact[];  // the analytical solution
psi_exact[left] = dirichlet(exact(x,y,0.));
psi_exact[right] = dirichlet(exact(x,y,0.));
psi_exact[top] = dirichlet(exact(x,y,0.));
psi_exact[bottom] = dirichlet(exact(x,y,0.));


int main() {

  FILE * fpglobal = fopen("error","w");

  L0=1.;
  
  int mylev=6;
  int Nobj = pow(2,mylev*2);  // initial objective number of element

  init_grid (1 << (mylev-1));

  scalar rhs[]; // RHS term for Poisson-Helmholtz equation

We do a loop to obtain several adapted meshes having N_{obj} elements.

For each N_{obj}, we do a loop to obtain the objective number of elements.

  for (int j=0; Nobj <= pow(2,2*8)/1.5; j++){

    for (int ki=0; ( fabs((double)(grid->tn - Nobj)/Nobj)) > 0.03 ; ki++){

      foreach ()
        rhs[] = src(x,y,0);

      poisson (psi, rhs, alp, lam, tolerance = TOL);   // poisson solver

We update an epsilon criterion to obtain the objective number of elements

      // AMR criterion 
      if (ki < 5 && j == 0) {  // estimate AMReps the first run with uniform refinement
        struct PreFactorData cd = compute_prefactors (2, {psi} );
        AMReps =  cd.cuniform*pow(Nobj,-1)/pow(Nobj,1./2.); 
      } else {
        AMReps = update_epsilon_control(Nobj);
      }

We adapt the mesh with an additionnal constraint on the minimal size computed from an estimation of the optimal conmpression ratio.

      /* restriction on the minimum grid size */
      double etaopt = estimate_eta_opt(2, {psi});
      maxlevel = 0.5*log(Nobj/etaopt)/log(2.);
            
      adapt_metric( {psi} );
      /* astats st = adapt_metric( {psi} ); */
    }

We compute total error and interpolation error.

    foreach ()
      rhs[] = src(x,y,0);
    poisson (psi, rhs, alp, lam, tolerance = TOL);

    // TOTAL ERROR
    double errtot = norm_gauss_5p (psi, user_norm, exact);  // compute total error :   ||u_num - u_exact||

    // INTERPOLATION ERROR estimate with metric-based formulae
    double Interr = Interpolation_error(user_norm);

    // INTERPOLATION ERROR exact
    foreach ()
      psi_exact[] = exact(x,y,0);

    double Int_err_exact = norm_gauss_5p (psi_exact, user_norm, exact);   // compute exact interpolation error :   ||u_interp - u_exact||

    // theoretical optimal and uniform errors
    struct PreFactorData cd;
    cd = compute_prefactors (2, {psi}, NULL );

    fprintf(fpglobal,"%ld %g %.10g %.10g %.10g %g\n", grid->tn, errtot, Int_err_exact, Interr, cd.copt*pow(grid->tn,-1), cd.cuniform*pow(grid->tn,-1));

We plot the total error and the interpolation error and we compare with the case without constraint. We plot the error in function of the square root of the number of element (equivalent to mean cell size in 2D). Both interpolation and total error are slightly higher with the constraint. This is logical for this case, as the total error is correlated with the interpolation error.

reset
set term pngcairo enhanced size 700,400
set output 'error.png'

set logscale
set xtics (16,32,64,128,256)
set format y "10^{%T}"
set xrange [64:190]

set key below
set multiplot layout 1,2
set title "total error"
p "error" u (sqrt($1)):6 w l t "uniform",\
"error" u (sqrt($1)):5 w l t "optimal",\
"error" u (sqrt($1)):2 w p t "total error with constraint",\
"./../../metric_amr_no_constraint/convergence_sharp/error" u (sqrt($1)):2 w p t "total error no constraint"

set title  "interpolation error"
p "error" u (sqrt($1)):6 w l t "uniform",\
"error" u (sqrt($1)):5 w l t "optimal",\
"error" u (sqrt($1)):3 w p t "interp error with constraint",\
"./../../metric_amr_no_constraint/convergence_sharp/error" u (sqrt($1)):3 w p t "interp error no constraint"

unset multiplot

Error (script)

We verify that we obtained the expected result

    assert ( fabs((double)(grid->tn - Nobj)/Nobj) <= 0.03);
    if (sqrt(Nobj)>150)
      assert(errtot<3.*cd.copt*pow(grid->tn,-(2./3.)) );

We ouput the mesh to compare with the computation with constraint

    scalar lev[];
    foreach()
      lev[]=level;

    char fname[80];
    sprintf(fname,"mesh_%i",Nobj);
    FILE * fp = fopen(fname,"w");
    output_field ({lev}, fp, n = 1 << 8, linear=false);
    fclose(fp);

loop update

    Nobj *= 1.5;

  }

  free_grid();
  fclose(fpglobal);
 
}

We compare the meshes with and without constraint. They are different.

reset
set term pngcairo enhanced size 700,700 font ",8"
set output 'mesh.png'

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set cbrange [4:10]

unset key
set size ratio -1
set pm3d map



set multiplot layout 1,2
set title  "with constraint"
sp "mesh_20736" u 1:2:3
set title  "no constraint"
sp "./../../no_hmin/sharp/mesh_20736" u 1:2:3
unset multiplot

Example of mesh which minimizes the interpolation error (script)