Crank-nicolson Finite Element Discretizations for a 2d Linear Schrödinger-type Equation Posed in a Noncylindrical Domain
نویسنده
چکیده
Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography that appears as an important application in Underwater Acoustics, we analyze a general Schrödinger-type equation posed on two-dimensional variable domains with mixed boundary conditions. The resulting initialand boundary-value problem is transformed into an equivalent one posed on a rectangular domain and is approximated by fully discrete, L-stable, finite element, Crank–Nicolson type schemes. We prove a global elliptic regularity theorem for complex elliptic boundary value problems with mixed conditions and derive L-error estimates of optimal order. Numerical experiments are presented which verify the optimal rate of convergence.
منابع مشابه
Crank-Nicolson finite element discretizations for a two-dimensional linear Schrödinger-type equation posed in a noncylindrical domain
Motivated by the paraxial narrow–angle approximation of the Helmholtz equation in domains of variable topography, we consider an initialand boundaryvalue problem for a general Schrödinger-type equation posed on a two space dimensional noncylindrical domain with mixed boundary conditions. The problem is transformed into an equivalent one posed on a rectangular domain and we approximate its solut...
متن کاملFinite Element Methods for Parabolic Equations
The initial-boundary value problem for a linear parabolic equation with the Dirichlet boundary condition is solved approximately by applying the finite element discretization in the space dimension and three types of finite-difference discretizations in time: the backward, the Crank-Nicolson and the Calahan discretization. New error bounds are derived.
متن کاملOn Fully Discrete Galerkin Methods of Second–order Temporal Accuracy for the Nonlinear Schrödinger Equation
We approximate the solutions of an initialand boundary-value problem for nonlinear Schrödinger equations (with emphasis on the ‘cubic’ nonlinearity) by two fully discrete finite element schemes based on the standard Galerkin method in space and two implicit, Crank–Nicolson-type second-order accurate temporal discretizations. For both schemes we study the existence and uniqueness of their soluti...
متن کاملOn Error Estimates of the Crank-Nicolson-Polylinear Finite Element Method with the Discrete TBC for the Generalized Schrödinger Equation in an Unbounded Parallelepiped
We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded n–dimensional parallelepiped (n ≥ 1). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transparent boundary conditions is considered. We present its stability properties and derive new error e...
متن کاملDiscrete Transparent Boundary Conditions for General Schrödinger–Type Equations
Transparent boundary conditions (TBCs) for general Schrödinger– type equations on a bounded domain can be derived explicitly under the assumption that the given potential V is constant on the exterior of that domain. In 1D these boundary conditions are non–local in time (of memory type). Existing discretizations of these TBCs have accuracy problems and render the overall Crank–Nicolson finite d...
متن کامل