// fem_check22_la.java  Galerkin FEM
// solve uxx+2*uxy+3*uyy+4*ux+5*uy+6*u=f(x,y
//       f(x,y) = 6x^3+6y^3+12x^2+15y^2+6xy+11x+22y+8
//       Analytic solution u(x,y)=x^3 + y^3 + xy + 1");
//       inside 4 by 4 grid, Gauss Legendre integration

class fem_check22_la
{
  // i, j, nx, ii, jj, ny, xg, yg needed by galk or galf, available at call 
  final int nx = 4;
  final int ny = 4;
  final int nxy = nx*ny;
  double xg[] = new double[nx];
  double yg[] = new double[ny];
  int i, j, ii, jj;
  laphi L = new laphi();

  fem_check22_la()
  {
    double x, y, hx, hy;
    final int npx = 6; // Gauss Legendre integration order
    final int npy = 12;
    double xx[] = new double[npx+1];
    double wx[] = new double[npx+1]; // for Gauss-Legendre 
    double yy[] = new double[npy+1];
    double wy[] = new double[npy+1]; // for Gauss-Legendre 
    double val, err, avgerr, maxerr;
    int px, py;

    double xmin = 0.0; // problem parameters 
    double xmax = 1.0;
    double ymin = 0.0;
    double ymax = 1.0;

    double kg[][] = new double[nxy][nxy];  
    double fg[] = new double[nxy];
    double ug[] = new double[nxy];
    double Ua[] = new double[nxy];


    System.out.println("fem_check22_la.java running  ");
    System.out.println("Given:");
    System.out.println("uxx+2*uxy+3*uyy+4*ux+5*uy+6*u=f(x,y)");
    System.out.println("f(x,y)=6x^3+6y^3+12x^2+15y^2+6xy+11x+22y+8");
    System.out.println("xmin<=x<=xmax ymin<=y<=ymax Boundaries  ");
    System.out.println("Analytic solution u(x,y)=x^3 + y^3 + xy + 1");

    System.out.println("xmin="+xmin+", xmax="+xmax+", ymin="+ymin+
                       ", ymax="+ymax);
    System.out.println("nx="+nx+", ny="+ny);
    System.out.println("x grid and analytic solution at ymin  ");
    hx = (xmax-xmin)/(double)(nx-1);
    for(i=0; i<nx; i++)
    {
      xg[i] = xmin+i*hx;
      Ua[i] = uana(xg[i],ymin); // temp 
      System.out.println("i="+i+", Ua("+xg[i]+")="+Ua[i]);
    }  
    System.out.println(" ");
    System.out.println("y grid and analytic solution at xmin ");
    hy = (ymax-ymin)/(double)(ny-1);
    for(ii=0; ii<ny; ii++)
    {
      yg[ii] = ymin+ii*hy;
      Ua[ii] = uana(xmin, yg[ii]); // temp 
      System.out.println("ii="+ii+", Ua("+yg[ii]+")="+Ua[ii]);
    }  
    System.out.println(" ");
    // put solution in Ua vector 
    for(i=0; i<nx; i++)
    {
      x = xmin+(double)i*hx;
      for(ii=0; ii<ny; ii++)
      {
	y = ymin+(double)ii*hy;
        Ua[i*ny+ii] = uana(x,y);
        System.out.println("solution at i="+i+",x="+x+", ii="+ii+",y="+y+
                           " is"+Ua[i*ny+ii]);
      }
    }
    System.out.println(" ");

    // initialize k and f 
    for(i=0; i<nx; i++)
    {
      for(j=0; j<nx; j++)
      {
        for(ii=0; ii<ny; ii++)
        {
          for(jj=0; jj<ny; jj++)
          {
            kg[(i*ny+ii)][(j*ny+jj)] = 0.0;
	  }
        }
      }
    }
    for(i=0; i<nx; i++)
    {
      for(ii=0; ii<ny; ii++)
      {
        fg[i*ny+ii] = 0.0;
      }
    }
    // set boundary conditions, four sides 
    for(i=0; i<nx; i++)
    {
      x = xg[i];
      ii=0;
      y = yg[ii];
      kg[(i*ny+ii)][(i*ny+ii)] = 1.0;
      fg[i*ny+ii] = uana(x,y);
      System.out.println("boundary i="+i+",x="+x+", ii="+ii+",y="+y+
                         " is "+fg[i*ny+ii]);
      ii=ny-1;
      y = yg[ii];
      kg[(i*ny+ii)][(i*ny+ii)] = 1.0;
      fg[i*ny+ii] = uana(x,y);
      System.out.println("boundary i="+i+",x="+x+", ii="+ii+",y="+y+
                         " is "+fg[i*ny+ii]);
    }
    for(ii=0; ii<ny; ii++)
    {
      y = yg[ii];
      i=0;
      x = xg[i];
      kg[(i*ny+ii)][(i*ny+ii)] = 1.0;
      fg[i*ny+ii] = uana(x,y);
      System.out.println("boundary i="+i+",x="+x+", ii="+ii+",y="+y+
                         " is "+fg[i*ny+ii]);
      i=nx-1;
      x = xg[i];
      kg[(i*ny+ii)][(i*ny+ii)] = 1.0;
      fg[i*ny+ii] = uana(x,y);
      System.out.println("boundary i="+i+",x="+x+", ii="+ii+",y="+y+
                         " is "+fg[i*ny+ii]);
    }

    System.out.println("calling gauleg xmin="+xmin+", xmax="+xmax+
                       ", npx="+npx);
    new gaulegf(xmin, xmax, xx, wx, npx); // xx and wx set up for integrations 
    System.out.println("xx[1]="+xx[1]+", xx[2]="+xx[2]+
                       ", wx[1]="+wx[1]+", wx[2]="+wx[2]);
    System.out.println("calling gauleg ymin="+ymin+", ymax="+ymax+
                       ", npy="+npy);
    new gaulegf(ymin, ymax, yy, wy, npy); // yy and wy set up for integrations 
    System.out.println("yy[1]="+yy[1]+", yy[2]="+yy[2]+
                       ", wy[1]="+wy[1]+", wy[2]="+wy[2]);
    i=1; ii=1;  j=1; jj=1;
    val = galk(xx[2],yy[2]);
    System.out.println("galk(xx[2],yy[2])="+val);
    val = galf(xx[2],yy[2]);
    System.out.println("galf(xx[2],yy[2])="+val);


    // compute entries in stiffness matrix 
    System.out.println("compute stiffness matrix  ");
    for(i=1; i<nx-1; i++)
    {
      for(j=0; j<nx; j++)
      {
        for(ii=1; ii<ny-1; ii++)
        {
          for(jj=0; jj<ny; jj++)
          {
            // compute integral for k(i,j)  
            val = 0.0;
            for(px=1; px<=npx; px++)
            {
              for(py=1; py<=npy; py++)
              {
                val = val + wx[px]*wy[py]*galk(xx[px],yy[py]);
	      }
            }
            System.out.println("Legendre integration="+val+", at i="+i+
                               ", j="+j+", ii="+ii+", jj="+jj);
            kg[(i*ny+ii)][(j*ny+jj)] = val;
          } // end jj loop 
        } // end ii loop 
      } // end j loop 
    } // end i loop 

    // compute integral for f(i)  
    for(i=1; i<nx-1; i++)
    {
      for(ii=1; ii<ny-1; ii++)
      {
        val = 0.0;
        for(px=1; px<=npx; px++)
        {
          for(py=1; py<=npy; py++)
          {
            val = val + wx[px]*wy[py]*galf(xx[px],yy[py]);
          }
        }
        System.out.println("Legendre integration="+val+", f at i="+i+
                           ", ii="+ii);
        fg[i*ny+ii] = val;
      }
    }
    System.out.println("k computed stiffness matrix, see above  ");
    System.out.println("f computed forcing function, see above  ");
    new simeq(kg, fg, ug);

    avgerr = 0.0;
    maxerr = 0.0;
    System.out.println("ug computed Galerkin, Ua analytic, error  ");
    for(i=0; i<nx; i++)
    {
      for(ii=0; ii<ny; ii++)
      {
        err = Math.abs(ug[i*ny+ii]-Ua[i*ny+ii]);
        if(err>maxerr) maxerr = err;
        avgerr = avgerr + err;
        System.out.println("ug["+i+","+ii+"]="+ug[i*ny+ii]+", Ua="+Ua[i*ny+ii]+
                           ", err="+(ug[i*ny+ii]-Ua[i*ny+ii]));
      }
    }
    System.out.println("                  maxerr="+maxerr+", avgerr="+
                       avgerr/(double)(nx*ny));
    System.out.println(" ");
  } // end fem_check22_la constructor

    // PDE problem definition functions
    double F(double x, double y)
    {
      return 6.0*x*x*x+6.0*y*y*y+12.0*x*x+15.0*y*y+6.0*x*y+11.0*x+22.0*y+8.0;
    }

    double uana(double x, double y) // u 
    {
      return x*x*x+y*y*y+x*y+1;
    }

    double galk(double x, double y)
    {                        // Galerkin k stiffness function for this problem 
      return (      L.phipp(x,j,nx-1,xg)*  L.phi(y,jj,ny-1,yg) +
              2.0 *  L.phip(x,j,nx-1,xg)* L.phip(y,jj,ny-1,yg) +
              3.0 *   L.phi(x,j,nx-1,xg)*L.phipp(y,jj,ny-1,yg) +
              4.0 *  L.phip(x,j,nx-1,xg)*  L.phi(y,jj,ny-1,yg) +
              5.0 *   L.phi(x,j,nx-1,xg)* L.phip(y,jj,ny-1,yg) +
              6.0 *   L.phi(x,j,nx-1,xg)*  L.phi(y,jj,ny-1,yg) )*
                      L.phi(x,i,nx-1,xg)*  L.phi(y,ii,ny-1,yg);
    } // end galk used by integration 

    double galf(double x, double y) // Galerkin f function for this problem 
    {
      return F(x,y)*L.phi(x,i,nx-1,xg)*L.phi(y,ii,ny-1,yg);
    } // end galf used by integration 

  public static void main (String[] args)
  {
    new fem_check22_la();
  }
} // end class fem_check22_la