We study one-dimensional wave equations defined by a class of fractal
Laplacians. These Laplacians are defined by fractal measures generated by
iterated function systems with overlaps, such as the well-known infinite
Bernoulli convolution associated with the golden ratio and the 3-fold
convolution of the Cantor measure. The iterated function systems defining these
measures do not satisfy the post-critically finite condition or the open set
condition. By using second-order self-similar identities introduced by
Strichartz et al., we discretize the equations and use the finite element and
central difference methods to obtain numerical approximations to the weak
solutions. We prove that the numerical solutions converge to the weak solution,
and obtain estimates for the rate of convergence
