Finite Difference Method Error Estimation

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L2 Norm

Suppose I am solving a pde with a solution known with a finite-difference method. I can represent it as $A_hu_h=f$ for some approximating matrix $A_h$. And I define the discrete norm in which I will analyze the convergence to be $$||e||^2=h\sum e_i^2$$. I have the estimate for the error $$||e||\leq C ||\tau||$$ for $\tau$ local error if first order in space and $C$ a stability bound for $A^{-1}$. Then, I want to see which error gives my program to confirm the theory. I pick a point on the grid and watch what happens as a double the mesh. Assume I see the error gets decreased by 2, so I suspect it is a linear convergence at that point. Thus, I can see only rate of convergence at one point. How can I guarantee that the convergence is uniform "everywhere" on the grid? to confirm the theory results, should not I measure the error at each point and see I have a linear convergence in order to claim that the error goes to zero linearly in a discrete norm defined above? Because I could state results in a similar norms such as discrete $L^{\infty}$ or discrete $L^1$, however, what I measure by the computer is the same: difference between numerical solution and the function value at the point. What confuses me is the fact that before I implement the method, I can state error estimates in a variety of norms, however, how do I relate these theoretical approximations to the error that I can actually measure, since this is just a value at the point on the grid and is independent from the way I do my theoretical analysis? Edit: I should rephrase the question: There are a number of papers, where the convergence is measured for a particular point on the grid and the results are stated to confirm the theoretical estimates that are done in discrete $L^{\infty}$ or $L^{2}$ for the error vector. I

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of error estimates for finite-difference methodsAuthorsAuthors and affiliationsM. N. SpijkerArticleReceived: 01 January 1971DOI: 10.1007/BF01398460Cite this article as: Spijker, M.N. Numer. Math. (1971) 18: 73. doi:10.1007/BF01398460AbstractIn this paper we study in http://link.springer.com/article/10.1007/BF01398460 an abstract setting the structure of estimates for the global (accumulated) error in semilinear finite-difference methods. We derive error estimates, which are the most refined ones (in a sense specified precisely in this paper) that are possible for the difference methods considered. Applications and (numerical) examples are presented in the following fields: 1. Numerical solution of ordinary as well as partial l2 error differential equations with prescribed initial or boundary values. 2. Accumulation of local round-off error as well as of local discretization error. 3. The problem of fixing which methods out of a given class of finite-difference methods are “most stable”. 4. The construction of finite-difference methods which are convergent but not consistent with respect to a given differential equation.References1.Ceschino, F., Kuntzmann, finite difference method J.: Problèmes différentiels de conditions initiales. Paris: Dunod 1963.Google Scholar2.Forsythe, G. E., Wasow, W. R.: Finite-difference methods for partial differential equations. New York: J. Wiley & Sons 1960.Google Scholar3.Godunov, S. K., Ryabenki, V. S.: Theory of difference schemes. Amsterdam: North-Holland Publishing Company 1964.Google Scholar4.Gragg, W. B., Stetter, H. J.: Generalized multistep predictor-corrector methods. J. Assoc. Comput. Mach.11, 188–209 (1964).Google Scholar5.Henrici, P.: Discrete variable methods in ordinary differential equations. New York: J. Wiley & Sons 1962.Google Scholar6.Hull, T. E., Luxemburg, W. A. J.: Numerical methods and existence theorems for ordinary differential equations. Numer. Math.2, 30–41 (1960).Google Scholar7.Isaacson, E., Keller, H. B.: Analysis of numerical methods. New York: J. Wiley & Sons 1966.Google Scholar8.Lees, M.: Discrete methods for nonlinear two-point boundary value problems. In: Numerical solution of partial differential equations, e. d. J. H. Bramble. New York: Academic Press 1966.Google Scholar9.Metté, A.: Essai de résolution du problème de Goursat par la methode de Runge-Kutta pour une equation aux dérivées partielles du type hyperbolique. Rev. Francaise Informat. Recherche Opérationnelle1, 67–90 (1967).Google Scholar10.Spijker, M. N.: Convergence and stability of step-by-step methods for the