Mimetic Discretizations of Continuum Mechanics

Stanly Steinberg

Problems in continuum mechanics are commonly described by initial boundary value problems for a system of partial differential equations.  Such problems can be discretized using finite difference, finite element, spectral, or many related techniques.  Mimetic methods follow a different route: they are not used to discretize particular  systems of equations, but rather to discretize the continuum theory.  Vector calculus provides a powerful invariant (coordinate-free) description of continuum mechanics as does the theory of  differential forms. In the vector calculus case, the operators gradient, curl and divergence play a central role: the equation of continuum mechanics can be written in terms of these operators along with the time derivative.  Thus mimetic methods for vector calculus provide discretizations of the gradient, curl and divergence, and then these discretizations are used to discretize the partial differential equations that appear in continuum mechanics problems.

Now the above doesn't make a mimetic discretization of continuum mechanics! The ideas of having a mimetic discretization is that if one can prove something about a particular continuum mechanics problem, for example the conservation of energy, then one should be able to prove the same thing in the discrete case. If one reviews the most elementary calculations for vector continuum mechanics, then the following list of items jump out as important.

First, in the continuum, the curl of the gradient and the divergence of the curl are zero, so this must also hold exactly in the discrete case. Not only that, but if the gradient of a scalar field is zero, then the field is constant, so this must hold in the discrete case also (this will eliminate many spurious mode problems). If the curl of a vector field is zero, then the vector field is the gradient of a scalar field, and if the divergence of a vector field is a zero, then it is the curl of another vector field, so these too must hold in the discrete case. The most general form of this type of result says that a vector field can be decomposed into a sum of the gradient of a scalar field and the curl of a vector field.  This result is important in some more advanced calculations.  For more information on this see the paper Natural Discretizations for the Divergence, Gradient, and Curl, by J. M. Hyman and M. Shashkov (1997) and their succeeding papers on this topic.

But the above is not enough.  Line, surface, and volume integrals play an important role in vector calculus,  so the discrete theory must have analogs of such integrals.  Once one has integrals, then one must have exact analogs of the integral theorems such as the divergence theorem.  At this point one will have enough tools to prove exact discrete conservation laws for the discrete analogs of systems that have conservation laws in the continuum.

But still this is not enough!  In linear problems the Lax Equivalence Theorem tells us that, for discrete systems, convergence to the continuum as the discretization is refined is equivalent to the uniform stability (with respects to the grids) of the discretization and the consistency of the discretization (the discretization error goes to zero as the grid is refined).  The existence of a conservation law implies the stability of the discretization, so one is left with estimating the accuracy of the approximation.

The above cited papers along with the paper Convergence of Mimetic Discretizations by J. M. Hyman, M. Shashkov,and S. Steinberg imply that you can, in fact, create such discretization for non-trivial problems.   Computational experience has show that these discretizations have excellent properties for problems where the grids are not smooth or the material properties are inhomogeneous or anisotropic.

The reader familiar with differential forms will realize that all said above can be stated more succinctly using this theory.