GS534: Solving the 1D Poisson equation using finite differences ...

⎡ (∆x) 2 ⎤
⎢
(∆x)
⎢
2 ⎥⎥⎥⎥⎥⎦
f = ⎢ (∆x) 2
(16c)
⎢ (∆x) 2
⎢ 1
⎣ 2 (∆x)2
Note that in this case N is the number of unknowns so that it doesn’t include the first nodal point
(for which the value is prescribed). N is therefore the number of nodal points minus one.
4.1. Solution of the discrete system: Gaussian elimination
The matrix vector system can be solved using various methods. A brute force method is to compute
the inverse A −1 of the matrix and find the solution by matrix-vector multiplication
u = A −1 f
(17)
It is generally quite expensive to compute the exact inverse of a matrix. One exception is for the
inversion of a diagonal matrix which is trivial. A more efficient method to solve (15) is to use
Gaussian elimination which generally requires O(N 3 ) operations for a full matrix but is significantly
more efficient for the sparse matrices that occur in finite difference methods. A general
approach is to decompose the matrix A into a upper and lower triangular system A = LU which
can be solved efficiently by substitution. The efficiency and memory requirements of Gaussian
elimination is much improved for special forms of matrix A. For example, when A is symmetric it
can be stored in approximately half the amount of memory and a more efficient LDL T where D is
a diagonal matrix. If the matrix is symmetric and positive definite one can compute the even
more efficient Cholesky decomposition LL T (see e.g., Golub and Van Loan, 1989 or Chapter 2 or
Press et al., 1992).
4.2. Matrix storage
An advantage for matrices arriving from the discretization of differential equations by finite differences
or finite elements is that they are generally sparse, with non-zero coefficients only within
a certain band surrounding the diagonal. For the matrix (16a) we have bandwidth one and because
of symmetry we can store the matrix by just the (main) diagonal ([2, 2, . . . . , 2, 1]) and the diagonal
line of coefficients right above it ([−1, −1, ..., −1]). The storage requirements are then 2N
which compares quite favorably with N 2 for the full matrix. The algorithm for Gaussian elimination
for a full matrix has to be modified to make use of the different storage, but the coefficients
outside the band are not affected in any way. Coefficients inside the band may be zero after discretization,
but they need to be stored explicity since they may become non-zero (’the band gets
filled’) during the matrix decomposition. This is not relevant for the matrix (12) but becomes
important for matrices that result from 2D or 3D discretization. For example, the matrix that
results from the discretization of the 2D Poisson equation is depicted in Figure 2b. This follows
from a 2D extension of the stencil (9) on a equidistant grid with 5 grid points along the horizontal
axis (verify). The banded matrices resulting from the discretization in Figure 2 have bandwidth
B = 1 (1D) or B = 5 (2D example), where the bandwidth is defined as that value B for which all
coefficients a ij are zero if j > i + B. For non-symmetric matrices we can make the distinction
between upper and lower bandwidth. In general it is more efficient to minimize the bandwidth. If
for example, one has a 2D grid of 10x5 nodal points it is best to number the nodal points in the
vertical direction first. That will give a bandwidth of 5, compared to a bandwidth of 10 for
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