Page 282 - Numerical Analysis and Modelling in Geomechanics

ENRICO PRIOLO 263
Discretisation of the wave equation
In order to obtain the spectral-element approximation of equations (9.4–9.7), Ω
is decomposed into rectangular non-overlapping elements Ω , and on the
e
decomposition the trial functions ũ(x, t) and the weight functions (x) are defined
such that

(9.14)

where ũ and denote the restrictions to Ω of ũ and , respectively. According to
e
e
the Galerkin approach, the functions ũ and take the following form in the local
e
coordinate system:
(9.15)

where and are the grid values of the unknown solution and of the weight
functions, respectively. Using the approximating function spaces (9.14) to solve
equation (9.2), it follows that the two-dimensional wave propagation problem is
equivalent to finding ũ such that for all the following equations are satisfied in
e
each element Ω :
e
(9.16)

enforcing the continuity condition for the solution on the element boundaries,
and where a(•,•) and (•,•) are symmetric, bilinear forms computed according to
N
N
definitions (9.5–9.7) at the element level.
Using the definition of φ (ζ) given in equation (9.13), we can compute the
i
derivative matrix D =dφ (ζ )/dζ, and then the semidiscrete differential operator
i
ij
j
(9.17)

The expansions (9.15) are now applied to the terms of equation (9.16) and the
(e)
resulting elemental integrals are evaluated using the mapping Λ (x) and the
semidiscrete operator . Requiring that the variational equation be satisfied for all ,
the spectral element approximation of the original equation finally yields a set of
linear differential equations