  We present a new framework for expressing finite element methods on multiple
intersecting meshes: multimesh finite element methods. The framework enables
the use of separate meshes to discretize parts of a computational domain that
are naturally separate; such as the components of an engine, the domains of a
multiphysics problem, or solid bodies interacting under the influence of forces
from surrounding fluids or other physical fields. Such multimesh finite element
methods are particularly well suited to problems in which the computational
domain undergoes large deformations as a result of the relative motion of the
separate components of a multi-body system. In the present paper, we formulate
the multimesh finite element method for the Poisson equation. Numerical
examples demonstrate the optimal order convergence, the numerical robustness of
the formulation and implementation in the face of thin intersections and
rounding errors, as well as the applicability of the methodology.
  In the accompanying paper~\cite{mmfem-2}, we analyze the proposed method and
prove optimal order convergence and stability.
