rafavzqz
commented
1 month ago Dear Shreya, to understand what's going on, you can look at the a priori error estimates available in IGA. For a smooth solution $u$, the error estimate for the $L^2$ norm (the $H^1$ norm is similar): $ || u - u_h ||0 <= C h^{p+1} || u ||{H^p} $ where $h$ is the mesh size, and $p$ is the degree. In the right-hand side it appears the $H^p$ norm of the solution, which takes into account its derivatives up to order $p$. That's why a high order polynomial gives you a higher error. There is also the "constant" $C$, which depends on the degree and the regularity of the discrete space, and also on the derivatives of the parametrization. That's probably why the error for the square domain is smaller. Note also that, since you are computing the absolute error, the size of the domain will also affect much the error.
Two comments:
- In the example ex_laplace_iso_ring, GeoPDEs will use an isoparametric discretization. Even if you choose degree 1, it will force you to use a NURBS space of degree 2 in the azimuthal direction. You can check that in space.degree.
- Since the exact solution is a high order polynomial, you should probably increase the number of quadrature points to obtain a more accurate value of the error. The asymptotic behavior should be correct, though.
Best regards, Rafael
shreyachauhanto888
Dear sir, I was experimenting with the Poisson problem. I am using the code given in the GeoPDEs software having title EX_LAPLACE_ISO_RING. The geometry of the domain is 'geo_ring.txt'. I am taking the exact solution $u(x,y)=-(x^2+y^2-1)(x^2+y^2-4)xy^2$ and taking the right side force function $f$ accordingly. With polynomial degree 1, regularity 0, number of subdivision 9 and points of Gaussian quadrature 2, I have calculated the $H^1$ and $L^2$ errors and they are error h1 = 2.7536 error l2 = 0.0825 . Now, if I change the exact solution to $u(x,y)=(x^2+y^2-1)^4(x^2+y^2-4)^4x^4y^4$ and with the same discretization parameters, the $H^1$ and $L^2$ errors which I obtained are error h1 =63.2450 error l2 =1.9063. My question is why the error in second case is quite more than that of in the first case? Is it due to the use of higher order polynomials in exact solution? I have also tried the same code with 'geo_square.txt' geometry with exact solution $u(x,y)=x^4(1-x)^4y^4(1-y)^4$ and with the same discretization parameters and the errors obtained are error h1 = 6.6284e-06 error l2 =1.8345e-07, which is comparatively quite less then the errors obtained in ring shaped domain. So, is the increase in the error is affected due to the shape of the domain?
Thanks in advance.
Regards, Shreya.