# Estimate $L_2$ norm of a elliptic problem with unknown exact solution on finite element method

I have the elliptic problem $$-\Delta u = 1,\,\,\Omega\subset\mathbb{R}^2$$ with $u=0$ on $\partial\Omega,$ with $\Omega=[-1,1]^2\backslash([0,1]\times[-1,0])$ and I want to estimate the $L_2$ error of linear finite element method on triangular mesh, if $u_h$ is the finite element solution and $u$ is the exact solution (unknown), $$\|u-u_h\|^2_{L_{2}(\Omega)}=\int_{\Omega}|u-u_h|^2\,dx.$$

Ι though to solve this problem with quadratic finite elements and set $u=\tilde{u}_h,$ where $\tilde{u}_h$ is the solution of quadratic finite elements.

• I'd recommend you check out this paper by Jaime Peraire at MIT. His group does a lot of work with DG methods, but they also work on error estimation for FEM solutions. This one is for the Poisson equation, but they have generalized the method to other equations as well. epubs.siam.org/doi/abs/10.1137/S0036142903425045 – Tyler Olsen Apr 16 '17 at 12:15
• I don't understand your question. Are you looking for an a posteriori estimator of the $L_2$ error? I also don't understand why you mean by "set $u=\tilde u_h$? $u$ is the exact solution of the equation, you can't set it to anything. Or are you suggesting to estimate the error of the $P_1$ solution by comparing it with the $P_2$ solution? – Wolfgang Bangerth Apr 16 '17 at 23:52
• @WolfgangBangerth Yes, I try to estimate the error of $P_1$ solution by comparing it with the $P_2$ solution and I do not know how to do. My goal is to compare the true $L_2$ error with the a posteriori estimator of the $L_2$ error, i.e compare the convergence rate. – math_lover Apr 17 '17 at 13:37
• You have two options: Either you solve a problem for which you know the exact solution (search for "Method of Manufactured Solutions"), or you compute a very accurate approximation to the solution. In the latter case, you could for example solve the problem on a very fine mesh, and compare against that, or use a higher order method. On coarse meshes, or if the solution lacks regularity, just taking the $P_2$ solution may not be accurate enough. – Wolfgang Bangerth Apr 17 '17 at 16:29
• No -- the $L_2$ norm needs to appropriately weigh each node. Furthermore, if $\tilde u$ is from a larger space (finer mesh, higher polynomial degree), then the integral is not equivalent to a sum over nodes. It needs to be done using quadrature of appropriately high order on the finer of the two meshes. – Wolfgang Bangerth Apr 18 '17 at 3:18