Hence, we are unable to solve this problem for N larger than 1,024 via Gaussian elimination. By looking at this table, it can be concluded that Gaussian elimination runs out of memory for N = 2,048. The ratios between error norms for consecutive rows in the table tend to 4, which confirms that the finite di erence method for this problem is second-order convergent with errors behaving like h 2, as predicted by the finite di erence theory. The norms of the finite di erence errors clearly go to zero as the mesh resolution N increases. , 11, for mesh of size N × N the number of degrees of freedom (DOF) which is dimension of the linear system N 2 the norm of the finite di erence error the ratio of consecutive error norms and the observed wall clock time in HH:MM:SS. In the process of solving this problem, we create Table 3.1 (a), which lists the mesh resolution N, where N = 2 ν for ν = 1, 2, 3. Refer to Appendix C.1 for Matlab code use to solve the linear system. This can be easily performed in Matlab using kron command. To create matrix A, we make use of the Kronecker tensor product. We know that this is easiest approach for solving linear systems for the user of Matlab, but it may not necessarily be the best method for large systems. Let us begin solving this linear system via Gaussian elimination. Thus, we will print this ratio in the following tables in order to confirm convergence of the finite di erence method. For su ciently small h, we can then expect that the ratio of errors on consecutively refined meshes behaves like Specifically, the finite di erence theory predicts that the error will converge as ku − u hk L∞(Ω) ≤ C h 2, as h → 0, where C is a constant independent of h. Since the solution u is su ciently smooth, we expect the finite di erence error to decrease as N gets larger and h = N 1 +1 gets smaller. The finite di erence error is defined as the di erence between the true solution u and the approximation u h defined on the mesh points. One of the things to consider to confirm the convergence of the finite di erence method is the finite di erence error. These formulas are the basis for the code in the function setupA shown in Appendix C.1. And I is the N × N identity matrix, and each of the matrices in the sum can be computed by Kronecker products involving T and I, so that A = I T + T I.
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