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The time-dependent Richards equation can be reformulated as a nonlinear, possibly degenerate, parabolic problem in mixed form by applying the Kirchhoff transformation. A preliminary time integration yields the variational formulation. A numerical treatment of this problem using polygonal and polyhedral meshes is, then, feasible by applying the mixed virtual element method. In this setting, we study a semi-discrete and a fully-discrete virtual element approximation. The theoretical analysis shows that our virtual element formulations are well-posed and convergent, and optimal convergence rates for the approximation errors can be proved. Such theoretical results are confirmed and the accuracy is assessed by investigating the behavior of the method on a set of representative numerical experiments.

We review basic design principles underpinning the construction of the mimetic finite difference and a few finite volume and finite element schemes for mixed formulations of elliptic problems. For a class of low-order mixed-hybrid schemes, we show connections between these principles and prove that the consistency and stability conditions must lead to a member of the mimetic family of schemes regardless of the selected discretization framework. Finally, we give two examples of using flexibility of the mimetic framework: derivation of arbitrary-order schemes and inexpensive convergent schemes for nonlinear problems with small diffusion coefficients.

The mimetic discretization technology relies on a discrete vector and tensor calculus (DVTC) that deals with discrete fields and discrete operators. The DVTC makes it possible to reproduce (or mimic) fundamental identities of continuum calculus, such as kernels of operators (see Sect. 2.6) and the Helmholtz decomposition theorems (see Sect. 2.7), in the discrete framework. It also guarantees symmetry and positivity of discrete operators when these properties hold for the corresponding differential operators.

The mathematical models used to describe our understanding of physical processes become more sophisticated every decade. Thanks to the enormous growth of computational capabilities, modern computer simulations include dozens of coupled physical phenomena. This imposes new requirements on the underlying numerical models. In addition to be accurate approximations of the mathematical models, the best discrete models try to preserve or mimic other important properties of PDEs such as the conservation laws, symmetries, maximum principles, and asymptotic limits. The mimetic finite difference (MFD) method is one of the existing tools used by numerical analysts to design such discrete models.

In the previous chapter we described the mimetic inner products that are low-order approximations of classical L 2 products of continuum functions u and v: $${[{u_{h,P}},{v_{h,P}}]_{{S_h},P}} = \int\limits_P {uvdV + O({h_P})|P|,} $$ where u h ,P3,v h ,P are discrete mesh functions from a space S h: $${u_{h,\operatorname{P} }} = \Pi _{h,\operatorname{P} }^S(u),\quad {v_{h,\operatorname{P} }} = \Pi _{h,\operatorname{P} }^S(v).$$

In this chapter we consider two different linear problems in structural mechanics. The first one is the linear elasticity problem, which we study in both the displacement-pressure and the stress-displacement formulations. In both cases, we take a particular care in devising mimetic schemes that are also stable in the incompressible limit. As the second problem, we consider the bending of Reissner-Mindlin plates, which is a very popular problem in engineering applications.

A generalized polyhedron is a topological polyhedron, i.e. a solid defined by a bi-Lipschitz mapping of a polyhedron. The faces of a generalized polyhedron are usually non-planar (or curved) but edges may remain straight line segments. A generalized polyhedral mesh is a mesh containing generalized polyhedra. Such a mesh often appears in Lagrangian fluid flow simulations where the computational mesh moves with flow.

The incompressible Stokes problem for the vector field u and the scalar pressure field p is governed by the following equations: $$ - \operatorname{div} \left({v\user2{\varepsilon }(\user2{u})} \right) + \nabla p = \user2{b}\quad \quad \operatorname{in} \;\Omega ,$$ (8.1) $$\operatorname{div} \;\user2{u} = 0\quad \quad \operatorname{in} \;\Omega ,$$ (8.2) $$\user2{u} = {\user2{g}^D}\quad \quad \operatorname{on} \;{\Gamma ^D},$$ (8.3) $$v\user2{\varepsilon }\left(\user2{u} \right) = {\user2{g}^N}\quad \quad \operatorname{on} \;{\Gamma ^N},$$ (8.4) where v > 0 is the fluid viscosity, the vector-valued field b is the forcing term, the vector-valued fields g D and g N are the boundary data, and e(u) = (∇u + (∇u) T )/2 is the symmetric strain tensor. We refer the reader to Sect. 1.5.1 for a more detailed presentation of the Stokes problem.

The first goal of this chapter is to derive explicit formulas for inner product matrices M P , M F , M E , and M P that were formally introduced in Chap. 2. The second goal is to create foundations for the theoretical analysis of mimetic discretizations.

A major property of the solutions of elliptic problems is the existence of the maximum and minimum principles [ 190, 203]. The strongest form of the minimum principle states that solution p cannot have a minimum in O when the source term is nonnegative. More precisely, if p takes a minimum value at point x 0 ∈ O, then p is constant in O. This classical result, also known as Hopf’s lemma [ 203], has been proved for p ∈C 2(O) and locally uniformly positive definite tensor K.

The diffusion problem in a mixed form is governed by the following set of equations: $${\mathbf{u}} + \user1{K}\nabla p = 0\;\;\;\;\;\;\;{\text{in}}\;\Omega ,$$ (5.1) $${\text{div}}\,{\mathbf{u}} = b\;\;\;\;\;\;\;{\text{in}}\;\Omega ,$$ (5.2) $$p = {g^D}\;\;\;\;\;\;\;{\text{on}}\;{\Gamma ^D},$$ (5.3) $${\mathbf{u}} \cdot {\mathbf{n}} = - {g^N}\;\;\;\;{\text{on}}\;{\Gamma ^N},$$ (5.4) where the vector variable u represents the flux of the scalar unknown p. The unknown p may be a pressure, a temperature, or a flow density depending on the physical interpretation that we give to this mathematical model. The mixed form of the diffusion problem provides an opportunity for a better approximation of the flux and the exact satisfaction of balance condition (5.2), e.g., [ 90, 205, 208].

The diffusion problem in primal form for the scalar variable u is governed by the Poisson problem, which we rewrite from Sect. 1.4.1: $$- div\left( {K\nabla u} \right) = b\quad in\;\Omega ,$$ (6.1) $$u = {g^D}\quad {\Gamma ^D},$$ (6.2) $$\left( {K\nabla u} \right) \cdot n = {g^N}\quad on\;{\Gamma ^N}.$$ (6.3)

Let Ω be a bounded, open, polygonal (polyhedral) subset of ℝ d , d = 2,3, with the Lipschitz continuous boundary Γ. We consider the mimetic discretization of the advection-diffusion equation for the scalar field p introduced in Sect. 1.4.3: $$div({\beta _p} - K{\nabla _p}) = b\,\,\,\,\,in\,\,\,\Omega ,$$ (10.1) $$p = {g^D}\,\,\,on\,\,\Gamma ,$$ (10.2) where K is a bounded, measurable, symmetric and uniformly elliptic tensor, b ∈ L 2(Ω), g D ∈ H 1/2(Γ), β ∈ C 1(\(\bar \Omega \)) d is such that div β ≥ 0, cf. assumptions (H1)– (H3) in Sect. 1.4.1. The diffusive and the total fluxes are given by $$u = - K{\nabla _p}\,\,and\,\,\tilde u = u + {\beta _p}.$$ (10.3)