I would like to know what are the necessary and sufficient tests one has to perform in order to show the convergence of the algorithm. I have not found a good reference to state for that as I am interested in the quantity $\|u_h-u\|$ in some discrete norm as $h\to 0$. $u_h$ denotes a numerical solution on a grid with spacing $h$ and $u$ denotes the true solution. Assume there is no exact solution to compare to. I don't want to "manufacture" the solution for the pde as it assumes that there is a strong solution of the pde, which I do not necessarily have. Bottom line, solution is not strong on the entire domain+the boundary. But I have a numerical algorithm that solves that equation and I can observe the results. What are my techniques to judge the accuracy for that equation with the used algorithm?
One of the ways I propose is ASSUME: $$\| u_h - u \| \leq Ch^2$$. Then $$\| u_{2h} - u_h \| \leq \| u_{2h} - u\| + \| u - u_h \| = 5Ch^2.$$ Similarly $$\| u_{4h} - u_{2h} \| \leq 20Ch^2$$ Thus, since all these solution live on different finite-dimensional spaces I project the finer solution on a coarser grid, that is $\| u_{2h} - u_h \|$ is measured on the grid space with spacing $2h$ and $\| u_{4h} - u_{2h} \| $ is measured on the grid spacing of $4h$. The ratio of these two quantities would give me $4$ for the second order method. However, I STARTED with an assumption that the method is second order and therefore to see $4$ as a ratio is only necessary, but not sufficient to claim that the algorithm in fact is of second order. Thus, what are my options to show that the rate of the algorithm is in fact second order in this case?
EDIT: the equation I am solving is $$u_t=u_{xx}+1_{\{x\geq K\}}u_y,\; x\in [0,2K], y\in [0,2], t\in [0,T]$$ with initial data: $$u(0,x,y)=x-K/2, x\in[K/2,2K], y \in [0,1]$$ and and everywhere else the value is $0$. So the initial data is "rough", lacking even continuity, the second dimension has no diffusion and the coefficient in pde is discontinuous, as a result, i do not obtain any regularity in $y$ dimension at all. And I don't see how I can manufacture a solution that would exhibit such a shape at $t=0$. For the manufactured solution idea, I have to come up with a function that has $u_t,u_{xx},u_y$ and is "very close" to the initial data, but since it will be smooth apriori in order to take derivative it is still not a good approximation of the real function.