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{  GAUS2D.PDE  

 

 This test solves a 2D heat equation with a Gaussian solution and compares

 actual deviations from the exact solution with the error estimates made by  

 FlexPDE.

 The problem runs a set of ERRLIM levels and plots the history of the comparison.

}  

 

title '2D Accuracy Test - Gaussian'

 

variables

   u

 

select

   ngrid=5

   errlim = staged(1e-2, 1e-3, 1e-4, 1e-5)

 

definitions

   k = 1

   h = 0.1

   w = 0.2   ! gaussian width

   u0 = exp(-(x^2+y^2)/w^2)

   source = -(dxx(u0)+dyy(u0))

   uxx_exact = dxx(u0)

   RMS_error = sqrt(integral((u-u0)^2)/sqrt(integral(u0^2)))

   fx = -2*x*u0/w^2

   fy = -2*y*u0/w^2

equations

   U: div(K*grad(u)) + source = 0

 

boundaries

  Region 1

      start(-1,-1) natural(u)=-fy line to (1,-1)

      value(u)=u0 line to (1,1)

      natural(u)=fy line to (-1,1)

      value(u) = u0 line to close

 

monitors

  grid(x,y)

  contour(u)

 

plots

  grid(x,y)

  contour(u)

  elevation(u,u0) from(-1,0) to (1,0)

  elevation(u-u0) from(-1,0) to (1,0)

  elevation(dxx(u),uxx_exact) from(-1,0) to (1,0)

  elevation(dxx(u)+dyy(u),-source) from(-1,0) to (1,0)

  contour(dxx(u)) contour(dxy(u)) contour(dyy(u))

  contour(space_error(u))

  contour(u-u0)

 

histories

  history(RMS_error, errlim) LOG

 

end