We study the finite element approximation of the solid isotropic material with penalization (SIMP) model for the topology optimization problem of minimizing the compliance of a linearly elastic structure. To ensure the existence of a local minimizer to the infinite-dimensional problem, we consider two popular regularization methods: $W^{1,p}$-type penalty methods and density filtering. Previous results prove weak(-*) convergence in the space of the material distribution to a local minimizer of the infinite-dimensional problem. Notably, convergence was not guaranteed to \emph{all} the isolated local minimizers. In this work, we show that, for every isolated local or global minimizer, there exists a sequence of finite element local minimizers that strongly converges to the minimizer in the appropriate space. As a by-product, this ensures that there exists a sequence of unfiltered discretized material distributions that does not exhibit checkerboarding.

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