Geophysically accurate model of mantle convection requires a mesh with a width on the order of ~1 km. Traditional codes represent the systems of equation by setting up the associated matrix. However, at the scales we want to operate, even forming this matrix is hardly possible. Thus, we use the matrix-free framework HyTeG to create the geophysical model and verify it with numerical experiments while assessing the scalability of the framework and laying out the difficulties involved.