Solve eigenvalue PDE problem
[v,l] = pdeeig(model,c,a,d,r)
[v,l] = pdeeig(b,p,e,t,c,a,d,r)
[v,l] = pdeeig(K,B,M,r)
[v,l] = pdeeig(model,c,a,d,r)
produces
the solution to the FEM formulation of the scalar PDE eigenvalue problem
$$-\nabla \cdot (c\nabla u)+au=\lambda du\text{on}\Omega $$
or the system PDE eigenvalue problem
$$-\nabla \cdot (c\otimes \nabla u)+au=\lambda du\text{on}\Omega ,$$
with geometry, boundary conditions, and mesh specified in model
,
a PDEModel
object. See Solve Problems Using PDEModel Objects.
r
is a two-element vector,
indicating an interval on the real axis. (The left-hand side can be -Inf
.)
The algorithm returns all eigenvalues in this interval in l
,
up to a maximum of 99 eigenvalues.
v
is an eigenvector
matrix. For the scalar case each column in v
is
an eigenvector of solution values at the corresponding node points
from p
. For a system of dimension N with n_{p} node
points, the first n_{p} rows
of v
describe the first component of v,
the following n_{p} rows of v
describe
the second component of v, and so on. Thus, the
components of v are placed in blocks v
as N blocks
of node point rows.
Note: Eigenvectors are determined only up to multiple by a scalar, including a negative scalar. |
The eigenvalue PDE problem is a homogeneous problem, i.e., only boundary conditions where g = 0 and r = 0 can be used. The nonhomogeneous part is removed automatically.
The coefficients c
, a
, d
of
the PDE problem can be given in a wide variety of ways. In the context
of pdeeig
the coefficients cannot depend on u
nor t
,
the time. For a complete listing of all options, see Scalar PDE Coefficients and Coefficients for Systems of PDEs.
[v,l] = pdeeig(b,p,e,t,c,a,d,r)
solves the problem using a mesh described by p
, e
,
and t
, with boundary conditions given by b
.
b
describes the boundary conditions of the
PDE problem. For the recommended way of specifying boundary conditions,
see Specify Boundary Conditions Objects. For all methods
of specifying boundary conditions, see Forms of Boundary Condition Specification.
The geometry of the PDE problem is given by the mesh data p
, e
,
and t
. For details on the mesh data representation,
see Mesh Data.
[v,l] = pdeeig(K,B,M,r)
produces the solution to the generalized sparse matrix eigenvalue
problem
K u_{i} = λB´MBu_{i}
u = Bu_{i}
with Real(λ) in the interval in r.
In the standard case c and d are positive in the entire region. All eigenvalues are positive, and 0 is a good choice for a lower bound of the interval. The cases where either c or d is zero are discussed next.
If d = 0 in a subregion, the mass matrix M becomes singular. This does not cause any trouble, provided that c > 0 everywhere. The pencil (K,M) has a set of infinite eigenvalues.
If c = 0 in a subregion, the stiffness
matrix K
becomes singular, and the pencil (K,M)
has many zero eigenvalues. With an interval containing zero, pdeeig
goes
on for a very long time to find all the zero eigenvalues. Choose a
positive lower bound away from zero but below the smallest nonzero
eigenvalue.
If there is a region where both c = 0 and d = 0, we get a singular pencil. The whole eigenvalue problem is undetermined, and any value is equally plausible as an eigenvalue.
Some of the awkward cases are detected by pdeeig
.
If the shifted matrix is singular, another shift is attempted. If
the matrix with the new shift is still singular a good guess is that
the entire pencil (K,M) is singular.
If you try any problem not belonging to the standard case, you must use your knowledge of the original physical problem to interpret the results from the computation.