Superconvergence estimates of finite element methods for American options

Superconvergence analysis of conforming finite element method for nonlinear Schrödinger equation

“…With the similar argument as [25] , we can easily prove the following superconvergence results: In the present work, we obtain the superclose and superconvergence results in H 1 -norm through a new approache for B-E and C-N fully-discrete schemes, which improve the conclusions in [20] and [22] . It seems that these results have never be seen in the existing literature.…”

“…For the semi-discrete scheme, [20] applied the conforming bilinear FE to the LSE, and obtained the superclose and superconvergence results in H 1 -norm; [21] derived the same results as [20] for NLSE with the conforming linear right triangular FE by establishing the relationship between Ritz projection and the linear interpolation; [22] developled a new mixed FE scheme for NLSE, and deduced the superconvergence results for original variable in H 1 -norm and auxiliary variable in L 2 -norm, respectively. Nevertheless, it is not an easy task for the fully-discrete case.…”

“…Nevertheless, it is not an easy task for the fully-discrete case. For example, [20] only discussed the convergence result with L 2 -norm for C-N scheme; [22] deduced the superclose results of order O (h 3 2 + ( t ) 1 2 ) for B-E scheme and O (h 3 2 + ( t ) 3 2 ) for C-N scheme in ∂ m v ∂ t m (·, t ) , p = + ∞ .…”

Low order nonconforming finite element method for time-dependent nonlinear Schrödinger equation

“…Especially, the superconvergence analysis of FEMs for the Schrödinger equation have been studied successfully. For example, [12] used the conforming bilinear element to solve the LSE and obtained the superclose and superconvergence results in H 1 -norm for the semi-discrete scheme. [13] derived the same results as [12] for NLSE with the conforming linear triangular element by establishing the relationship between Ritz projection and the linear interpolation.…”

“…For example, [12] used the conforming bilinear element to solve the LSE and obtained the superclose and superconvergence results in H 1 -norm for the semi-discrete scheme. [13] derived the same results as [12] for NLSE with the conforming linear triangular element by establishing the relationship between Ritz projection and the linear interpolation. Whereafter, a series of superconvergence results about backward Euler and Crank-Nicolson fully-discrete schemes for NLSE also were studied in [14][15][16][17].…”