  We consider several mathematical issues regarding models that simulate forces
exerted by cells. Since the size of cells is much smaller than the size of the
domain of computation, one often considers point forces, modelled by Dirac
Delta distributions on boundary segments of cells. In the current paper, we
treat forces that are directed normal to the cell boundary and that are
directed toward the cell centre. Since it can be shown that there exists no
smooth solution, at least not in $H^1$ for solutions to the governing momentum
balance equation, we analyse the convergence and quality of the approximation.
Furthermore, the expected finite element problems that we get necessitate
scrutinizing alternative model formulations, such as the use of smoothed Dirac
distributions, or the so-called smoothed particle approach as well as the
so-called 'hole' approach where cellular forces are modelled through the use of
(natural) boundary conditions. In this paper, we investigate and attempt to
quantify the conditions for consistency between the various approaches. This
has resulted in error analyses in the $H^1$-norm of the numerical solution
based on Galerkin principles that entail Lagrangian basis functions. The paper
also addresses well-posedness in terms of existence and uniqueness. The current
analysis has been performed for the linear steady-state (hence neglecting
inertia and damping) momentum equations under the assumption of Hooke's law.
