1.1. Formulation of the problem
Let , with and , , and be the solution of the following initial and boundary value problem:
where with , and with
1.2. Formulation of the numerical method
Let be the set of all positive integers. For a given , we define a uniform partition of the time interval with time-step , nodes for , and intermediate nodes for . Also, for given , we consider a uniform partition of with mesh-width and nodes for , and a uniform partition of with mesh-width and nodes for . Also, we set , , , and introduce the discrete space
and a discrete Laplacian operator by
In addition, we introduce an operator , which, for given , is defined by for all . Finally, for any , we define by for all .
The Relaxation Finite Difference (RFD) method uses a standard finite difference scheme for space discretization along with a variant of the Relaxation Scheme for time stepping (cf. ).
Step 1: First, set
and find such that
Step 2: Set
and find such that
Step 3: For , first set
and then find such that
1.3. References and main results
The Relaxation Scheme (RS) has been introduced by C. Besse  as a linearly implicit, conservative, time stepping method for the approximation of the solution to the nonlinear Schrödinger equation. Convergence results has been obtained in  and  for the (RS) time-discrete approximations of the Cauchy problem for the nonlinear Schrödinger equation, which can not be concluded in the fully-discrete case, because on the one hand they are valid for small final time , and on the other hand are based on the derivation of a priori bounds (with ) for the time discrete approximations. Recently, in , the (RS) joined with a finite difference scheme is proposed for the approximation of the solution to a semilinear heat equation in the 1D case, and the corresponding error analysis arrived at optimal second order error estimates. Here, we investigate the extension of the results obtained in  in the 2D case. We would like to stress that a fylly-discrete version of the (RS) applied on a parabolic problem can be analyzed by using energy estimates, which, however, are not efficient in the case of the nonlinear Schrödinger equation (see ).
To develop an error analysis for the (RFD) method, we introduce the modified scheme (see Section 3.2) that follows from the (RFD) method after mollifying properly the terms with nonlinear structure (cf. , , ). For the approximations obtained from the modified scheme, we provide an optimal, second order error estimate in the discrete norm at the nodes and in the discrete norm at the intermediate nodes (see Theorem 3.1). After applying an inverse inequality (see (2.3)) and imposing a proper mesh condition (see (3.31)), the latter convergence result implies that the discrete norm of the modified approximations is uniformly bounded, and thus they are the same with those derived by the (RFD) method and hence inherit their convergence properties o (see Theorem 3.2), i.e. that there exist constant , independent of , , , such that
where is a discrete norm and is a discrete norm.
2.1. Discrete relations
We provide with the discrete inner product given by
and we shall denote by its induced norm, i.e. for . Also, we equip with a discrete -norm defined by for , and with a discrete -type norm given by
In the convergence analysis of the method, we will make use of the following, easy to verify, relation
of the discrete Poincaré-Friedrishs inequality
of the inverse inequality
For it holds that
and of the following Lipschitz-type inequality:
Let with . Then, for , it holds that
It is similar to the proof of (2.9) in Lemma 2.3 in . ∎
2.2. Consistency Errors
To simplify the notation, we set for , and for .
2.2.1. Consistency error in time
For , let be defined by
Then, setting and expanding, by using the Taylor formula around , we obtain:
which, easily, yields
2.2.2. Consistency error in space
3. Convergence Analysis
3.1. A mollifier
3.2. The modified scheme
Step M1: First, we set
Step M2: Set
and find such that
Step M3: For , first define by
and, then, find such that
3.3. Convergence of the modified scheme
Let and . Then, there exist positive constants , and , independent of and , such that: if , then
To simplify the notation, we set for , and for . In the sequel, we will use the symbol to denote a generic constant that is independent of , and , and may changes value from one line to the other. Also, we will use the symbol to denote a generic constant that depends on but is independent of , , and may changes value from one line to the other.