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IntegralTestFunctions.hpp

#ifndef _INTEGRALTESTFUNCTIONS_
# define _INTEGRALTESTFUNCTIONS_

template<int DIM>
struct IntegralTestFunction : public Function {
  virtual void   SetParameters(double *) {;}
  virtual double Eval(double *x)         { double q=1.0 ; for (int i=0; i<DIM; i++)  q *=x[i]*(1+x[i]*(-3+x[i]*2)) ; return q ;}
} ;

// 
// f(x)=exp(|x|^2 * alpha) 
struct IntegroDiffTestFunction : public Function {
  double alpha ;
  virtual void   SetParameters(double *p) {alpha=*p; }
  virtual double Eval(double *x)          {return exp((x[0]*x[0] + x[1]*x[1])*alpha); }  
} ;


// f(x)=prod_i x_i(1-x_i*x_i)
template<int DIM>
struct IntegroDiffTestKFunction : public Function {
  virtual void   SetParameters(double *) {;}
  virtual double Eval         (double *x){double q=1.0; for (int i=0; i<DIM; i++) q *= x[i]*(1-x[i]*x[i]); return q;}
} ;


// f(x)=\Delta exp(|x|^2 * alpha) 
template<int DIM>
struct IntegroDiffTestRHSFunction : public IntegroDiffTestKFunction<DIM> {
 virtual double Eval(double *x) { 
                  double r=x[0]*x[0] + x[1]*x[1], a=((exp(alpha)-1)/alpha -1)/(2*alpha) ;
                  return -4*alpha*exp(r*alpha)*(1+alpha*r) + 40000*a*a*x[0]*(1-x[0]*x[0])*x[1]*(1-x[1]*x[1]) ;
 }
} ;


// 
// u(x)=(1-x)(1-y)/(1 + (x-1/2)^2 + (y-1/2)^2)
struct IntegroDiffTest2Function : public Function {
  virtual void   SetParameters(double *) {;}
  virtual double Eval(double *xx) { 
    double x=xx[0],y=xx[1] ;
    return x*(1-x)*y*(1-y)/(1+20*((x-1./3)*(x-1./3) + (y-1./8)*(y-1./8))) ;
  }
} ;

//
// K(x1,x2 , y1,y2) = sin((1-x1)(1-x2)(1-y1)(1-y2)) * (1 + (y1-1/2)^2 + (y2-1/2)^2)  
struct IntegroDiffTestK2Function : public Function {
  virtual void   SetParameters(double *) {;}
  virtual double Eval(double *x) {
    double x1=x[0],x2=x[1],y1=x[2],y2=x[3] ;
    return sin( (1-x1)*(1-x2)*(1-y1)*(1-y2) ) * (1 + 20*((y1-1./3)*(y1-1./3) + (y2-1./8)*(y2-1./8))) ;
  }
} ;

// 
// f(x)=\Delta u + int_y K(x,y)u(y) dy
struct IntegroDiffTestRHS2Function : public Function {
  virtual void   SetParameters(double *) {;}
  virtual double Eval(double *xx) {
    double X =(1-xx[0])*(1-xx[1]),x=xx[0],y=xx[1], 
           x2=x*x, x3=x2*x, x4=x3*x,
           y2=y*y, y3=y2*y, y4=y3*y,
           X2=X*X, X3=X2*X, X4=X3*X, X5=X4*X, 
           n =(509 + 2880*x2 - 1920*x +2880*y2 - 720*y), 
           z =( 718199*y + 107399*x 
                +24883200*(x*y4 + x2*y3 -x2*y4 -y2*x4      -x*y3   +y*x4)
                +38707200*y2*x3 -12441600*y4*y
                + 8294400*x4*x2 -38707200*y*x3
                + 8294400*y4*y2 + 2728800*y3
                + 2067840*y4    -31605120*x2*y2
                +15603840*x4    -19353600*x4*x
                - 5116800*x3    +  464761*x2
                - 1367639*y2    +15016320*x*y2
                - 8795520*x*y   +25384320*x2*y), 
           k=(X/144 + X5/376320 +X4*X5/6322821120. 
               + X5*X5*X3/358676398080000. ) -
               (X3/5400 + X4*X3/40824000 + X5*X5*X/1322204083200.) ;

    return  -288*z/(n*n*n) + 1000*k ;
  }
} ;


// 
// f(x)=\Delta u + int_y K(x,y)u(y) dy
struct IntegroDiffTestRHSTest2Function : public Function {
  virtual void   SetParameters(double *) {;}
  virtual double Eval(double *xx) {
    double X=(1-xx[0])*(1-xx[1]) ;
    return (X/144 + X*X*X*X*X/376320 +X*X*X*X*X*X*X*X*X/6322821120. 
            + X*X*X*X*X*X*X*X*X*X*X*X*X/358676398080000. ) -
      (X*X*X/5400 + X*X*X*X*X*X*X/40824000 + X*X*X*X*X*X*X*X*X*X*X/1322204083200.) ;
  }  
} ;


#endif

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