Quickly, the

Mixed Finite Element method , a great reference is of course Brezzi and Fortin, deals with elliptic problems where the unknowns belong to different functional spaces. As an example, take the incompressible Stokes problem where the unknowns are the velocity and the pressure. For a Dirichlet boundary condition problem it is known that u belongs to H^1_0 and the pressure p is in L^2_0. Two different spaces.

For the Mixed FEM the choice of elements, one for u and another for p, is of paramount importance. The choice must satisfy a compatibility condition which known as the discret inf-sup condition.

The Discontinuous Galerkin FEM proceeds as the regular Galerkin FEM , meaning project the continuous problem into a finite dimensional space of dimension n and seek the numerical solution to your problem as a linear combination over the basis that spans the finite dimensional space. In regular FEM we impose continuity of the sought solution from one element to another, this condition is relaxed when we deal with the DGFEM. The sought solution is allowed to be discontinuous from one element to another.

The discontinuous Galerkin method is a robust and compact finite element projection method that provides a practical framework for the development of high-order accurate methods using unstructured grids. The method is well suited for large-scale time-dependent computations in which high accuracy is required. An important distinction between the discontinuous Galerkin method and the Mixed Finite-Element Method is that in the discontinuous Galerkin method the resulting equations are local to the generating element. The solution within each element is not reconstructed by looking to neighboring elements. Therefore, each element may be thought of as a separate entity that merely needs to obtain some boundary data from its neighbors. The compact form of the discontinuous Galerkin method makes it well suited for parallel computer platforms. This compactness also allows a heterogeneous treatment of problems. That is, the element topology, the degree of approximation and even the choice of governing equations can vary from element to element and in time over the course of a calculation without loss of rigor in the method.

Many of the method's accuracy and stability properties have been rigorously proven for arbitrary element shapes, any number of spatial dimensions, and even for nonlinear problems, which lead to a very robust method. The discontinuous Galerkin method has been shown in mesh refinement studies to be insensitive to the smoothness of the mesh. Its compact formulation can be applied near boundaries without special treatment, which greatly increases the robustness and accuracy of any boundary condition implementation. These features are crucial for the robust treatment of complex geometries. In semi-discrete form, the discontinuous Galerkin method can be combined with explicit time-marching methods, such as Runge-Kutta. One of the disadvantages of the method is its high storage and high computational requirements; however, a recently developed quadrature-free implementation has greatly ameliorated these concerns.

I hope I have answered your question and wish you the best of luck.

Mixed finite elements, for which a classical reference is

Brezzi, F., & Fortin, M. (2012). Mixed and hybrid finite element methods (Vol. 15). Springer Science & Business Media.

are usually characterized by the introduction of extra degrees of freedom for the fluxes of some quantities of interest. In some DG approaches, like the so called Local Discontinuous Galerkin approach, see e.g.

Cockburn, Bernardo, and Chi-Wang Shu. "The local discontinuous Galerkin method for time-dependent convection-diffusion systems." SIAM Journal on Numerical Analysis 35.6 (1998): 2440-2463.

Castillo, P., Cockburn, B., Perugia, I., & Schötzau, D. (2000). An a priori error analysis of the local discontinuous Galerkin method for elliptic problems. SIAM Journal on Numerical Analysis, 38(5), 1676-1706.

and many others, exactly the same procedure is followed. There are then many differences, since in standard mixed finite elements the mixed formulation is then

usually discretized by continuous elements, while the LDG approach employs

discontinuous finite element spaces and uses numerical fluxes to define uniquely boundary terms. Therefore, the mixed formulation of elliptic problems is used also in the DG context, even though standard mixed finite elements and DG yield different discretizations. I hope it helps....

Thank you, colleagues

Your considerations are perfectly academic and strict

May explanation was that in DG you don't have have shared DOF on edges of elements as you have in mixed FEM.

Consider simple diffusion problem that is reduced to equation for divergence of a flux vector.

In Mixed Fem according to Brezzi Fortin book you approximate the flux with RT0(for example) elements when a common edge has the same flux vector component for couple of neighbors.

In DG you approximate the flux vector separately inside each element even for neighbors with common edge.

Regards, Egor

You are welcome and good luck.

DG has two sets of DOFs along the cell/volume edge. The numerical flux is used to derive a uniquely defined value of the quantities of interest. Another approach that has become popular in the last decade and which has some kind of similarity to mixed FEM is Hybridizable DG methods, see e.g.

Nguyen, N. C., Peraire, J., & Cockburn, B. (2009). An implicit high-order hybridizable discontinuous Galerkin method for linear convection–diffusion equations. Journal of Computational Physics, 228(9), 3232-3254.

Mixed FEM are conforming discretization schemes based on the mixed formulation of the PDE. The natural functional spaces appearing in mixed formulations of elliptic problems are H(div) for the additional variable (flux/stress/ecc..) and L^2 for the primary variable (pressure/displacement/ecc..). In order to have conformity of the flux/stress, there is the need to enforce the continuity of the normal traces at the mesh skeleton (internal edges/faces). That's why the use of shared DOFs on edges/faces.

Discontinuous Galerkin methods yields nonconforming approximations of the problem and their design can either be based on primal or mixed weak formulation of the PDE as detailed in

Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems, D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini, (SINUM 2002 39:5, 1749-1779).

In dG methods the inter-element continuity requirements are usually imposed weakly by penalty terms. Indeed, discrete spaces are spanned by fully discontinuous piecewise polynomials on the mesh and there are no shared DOFs.

I hope that this answer gives you a clear (and quite short) explanation that also relate to your previous consideration.

Best regards!

Michele, thank you! People like me are getting used to forget Philippe Ciarlet's classic. A classification from his book works fine.

Philippe G. Ciarlet The Finite Element Method for Elliptic Problems , North Holland 1978

Regards!

Quickly, the

Mixed Finite Element method , a great reference is of course Brezzi and Fortin, deals with elliptic problems where the unknowns belong to different functional spaces. As an example, take the incompressible Stokes problem where the unknowns are the velocity and the pressure. For a Dirichlet boundary condition problem it is known that u belongs to H^1_0 and the pressure p is in L^2_0. Two different spaces.

For the Mixed FEM the choice of elements, one for u and another for p, is of paramount importance. The choice must satisfy a compatibility condition which known as the discret inf-sup condition.

The Discontinuous Galerkin FEM proceeds as the regular Galerkin FEM , meaning project the continuous problem into a finite dimensional space of dimension n and seek the numerical solution to your problem as a linear combination over the basis that spans the finite dimensional space. In regular FEM we impose continuity of the sought solution from one element to another, this condition is relaxed when we deal with the DGFEM. The sought solution is allowed to be discontinuous from one element to another.