Multi-grained parallel computers can be very effective on computationally intensive problems that have important se- rial and parallel components. We introduce the Mesh Su- per Het, a model of this type consisting of the close coupling of a d-dimensional toroidal mesh of coarse-grained proces- sors to a serial machine consisting of memory modules con- nected via a low-diameter network to a serial processor. We exhibit problems for which the Mesh Super Het is superior to its serial or parallel components alone and develop tight performance bounds for sorting, the fast Fourier transform, and matrix multiplication. As multi-grained machines be- come more common, studies such as this will both reveal the fundamental limitations on such architectures and set the context for algorithm development.