A FEniCS Tutorial - FEniCS Project

The object A is of type Matrix, while b and u.vector() are of type Vector.
We may convert the matrix and vector data to numpy arrays by calling the
array() method as shown before. If you wonder how essential boundary conditions
are incorporated in the linear system, you can print out A and b before
and after the bc.apply(A, b) call:
A = assemble(a)
b = assemble(L)
if mesh.num_cells() < 16: # print for small meshes only
print A.array()
print b.array()
bc.apply(A, b)
if mesh.num_cells() < 16:
print A.array()
print b.array()
WithaccesstotheelementsinAthroughanumpyarraywecaneasilyperform
computations on this matrix, such as computing the eigenvalues (using the
eig function in numpy.linalg). We can alternatively dump A and b to file in
MATLAB format and invoke MATLAB or Octave to analyze the linear system.
Dumping the arrays A and b to MATLAB format is done by
import scipy.io
scipy.io.savemat(’Ab.mat’, {’A’: A, ’b’: b})
Writing load Ab.mat in MATLAB or Octave will then make the variables A
and b available for computations.
Matrix processing in Python or MATLAB/Octave is only feasible for small
PDE problems since the numpy arrays or matrices in MATLAB file format are
densematrices. DOLFINalsohasaninterfacetotheeigensolverpackageSLEPc,
which is a preferred tool for computing the eigenvalues of large, sparse matrices
of the type encountered in PDE problems (see demo/la/eigenvalue in the
DOLFIN source code tree for a demo).
A complete code where the linear system AU = b is explicitly assembled and
solved is found in the file dn3_p2D.py in the directory stationary/poisson.
This code solves the same problem as in dn2_p2D.py (Section 1.14). For small
linear systems, the program writes out A and b before and after incorporation of
essential boundary conditions and illustrates the difference between assemble
and assemble_system. The reader is encouraged to run the code for a 2 × 1
mesh (UnitSquare(2, 1) and study the output of A.
By default, solve(A, U, b) applies sparse LU decomposition as solver.
Specification of an iterative solver and preconditioner is done through two optional
arguments:
solve(A, U, b, ’cg’, ’ilu’)
Appropriate names of solvers and preconditioners are found in Section 7.4.
To control tolerances in the stopping criterion and the maximum number of
iterations, one can explicitly form a KrylovSolver object and set items in its
parameters attribute (see also Section 1.5):
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