Benchmark computations for Schrödinger equations
with constant magnetic potential in the unit square
and Neumann boundary conditions

Contents

Problem   Task   Refs   Sols   Nums  

The Problem

The domain is the unit square $\Omega =$ (-1,1) x (-1,1). The coordinates are (x,y).

The operator is the positive hermitian operator $$ - \left(h\partial_x + i \frac{y}{2}\right)^2 - \left(h\partial_y - i \frac{x}{2}\right)^2 $$ depending on the (small) parameter h.
More explicitely, the bilinear form associated with this problem is $$ \eqalign { a(u,v) &= \int_\Omega \Big(h\partial_xu + \frac{iy}2 u\Big) \Big(h\partial_xv - \frac{iy}2 v \Big) + \Big(h\partial_yu - \frac{ix}2 u \Big) \Big(h\partial_yv + \frac{ix}2 v \Big) \ dxdy \cr & = h^2 \int_\Omega (\partial_xu\,\partial_xv+\partial_yu\,\partial_yv) \ dxdy \cr &\quad + ih \int_\Omega \left(\frac{y}2 (u\,\partial_xv - \partial_xu\,v) - \frac{x}2 ( u \,\partial_yv - \partial_yu \,v) \right)\ dxdy \cr &\quad+ \int_\Omega \frac14(x^2+y^2)\,u\,v \ dxdy } $$ defined on the variational space $H^1(\Omega)$.

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The Task         Back to Contents

Compute the first 8 eigenpairs for
    h=0.1
    h=0.07
    h=0.05
    h=0.03
    h=0.02
    h=0.014
    h=0.01

References         Back to Contents

Virginie Bonnaillie-Noel, Monique Dauge, Daniel Martin, Gregory Vial.
Computations of the first eigenpairs for the Schrödinger operator with magnetic field
Computer Methods in Applied Mechanics and Engineering 196 (2007) 3841-3858
../publis/ViMoDanGreg0512.html    HAL

Virginie Bonnaillie, Monique Dauge.
Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners
Ann. Henri Poincaré 7, (2006), 899–931.
../publis/ViMo0510.html    HAL

Visual solutions         Back to Contents

Computed with uniform p-version of degree 10 on a 8 x 8 grid.

View the real parts of the first 8 eignemodes, and their moduli.
Follow the links!

Classified versus h
    h = 0.1
    h = 0.07
    h = 0.05
    h = 0.03
    h = 0.02
    h = 0.014
    h = 0.01

Numerics         Back to Contents

Computed by p-version of degree 36 (and degree 40 for h=0.01) on a 2 x 2 mesh (4 elements!)
With the number of reasonably reliable digits.

h = 0.10 .477629673381910E-01
.482313382915062E-01
.575082202536651E-01
.580233919748928E-01
.828889137319983E-01
.878614372363941E-01
.957506570182964E-01
.120996144114729E+00 		10
10
10
10
10
10
10
10

h = 0.07 .339661904852459E-01
.348003789346719E-01
.374408920886324E-01
.388641867546694E-01
.529431293660633E-01
.532538400393957E-01
.614419262447093E-01
.658601048436502E-01 		10
10
10
10
10
10
10
10

h = 0.05 .247429551033885E-01
.250558770565737E-01
.261885311611044E-01
.266587594290357E-01
.351346240490286E-01
.351523746280324E-01
.395022774946217E-01
.412068817918855E-01 		10
10
10
10
10
10
10
10

h = 0.03 .150853442229727E-01
.152679212151558E-01
.153397571546128E-01
.155729836717143E-01
.193677349015526E-01
.198120602283544E-01
.208118545145875E-01
.213462243684591E-01 		10
10
10
10
10
10
10
10

h = 0.02 .101453242800229E-01
.101726792564451E-01
.102257479719875E-01
.102585122102338E-01
.124898074056269E-01
.127777032902530E-01
.128581119036340E-01
.133503546274100E-01 		10
10
10
10
10
10
10
10

h = 0.014 .712602059232523E-02
.712971355701103E-02
.714813593668865E-02
.715203727032689E-02
.861004925836663E-02
.864003851443441E-02
.881505595256245E-02
.888781316955205E-02 		10
10
10
10
10
10
10
10

h = 0.01 .509588376909173E-02
.509729219004108E-02
.510086928656686E-02
.510233168236355E-02
.606447246477928E-02
.611315479068834E-02
.612609845595872E-02
.620289037068536E-02 		6
6
6
6
6
5
5
5

 

Created in Banff Center (BIRS), June 28, 2012. Last modified: June 29, 2012