## Two classes of mixed finite element methods.(English)Zbl 0629.73053

Finite element methods are presented in an abstract setting for mixed variational formulations. The methods are constructed by adding to the classical Galerkin method various least-squares like terms. The additional terms involve integrals over element interiors, and include mesh-parameter dependent coefficients. The methods are designed to enhance stability. Consistency is achieved in the sense that exact solutions identically satisfy the variational equations. Applied to various problems, simple finite element interpolations are rendered convergent, including convenient equal-order interpolations which are generally unstable within the Galerkin approach.
The methods are subdivided into two classes according to the manner in which stability is attained: (1) Circumventing Babuška-Brezzi condition methods.
(2) Satisfying Babuška-Brezzi condition methods.
Convergence is established for each class of methods. Applications of the first class of methods to Stokes flow and compressible linear elasticity are presented. The second class of methods is applied to compressible and incompressible elasticity problems.

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74S30 Other numerical methods in solid mechanics (MSC2010) 65K10 Numerical optimization and variational techniques
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### References:

 [1] Arnold, D.N., Discretization of finite elements of a model parameter dependent problem, Numer. math., 37, 405-421, (1981) · Zbl 0446.73066 [2] Arnold, D.N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 4, 337-344, (1984) · Zbl 0593.76039 [3] Arnold, D.N.; Brezzi, F.; Douglas, J., PEERS: A new mixed finite element for plane elasticity, Japan J. appl. math., 1, 2, 347-367, (1984) · Zbl 0633.73074 [4] Arnold, D.N.; Douglas, J.; Gupta, C.P., A family of higher order mixed finite element methods for plane elasticity, Numer. math., 45, 1-22, (1984) · Zbl 0558.73066 [5] Arnold, D.N.; Falk, R.S., A new mixed formulation for elasticity, (1986), preprint [6] Babuška, I., Error bounds for finite element method, Numer. math., 16, 322-333, (1971) · Zbl 0214.42001 [7] Bercovier, M., Perturbation of mixed variational problems. application to mixed finite element methods, RAIRO anal. numér., 12, 3, 211-236, (1978) · Zbl 0428.65059 [8] Bercovier, M.; Pironneau, O., Error estimates for finite element method solution of the Stokes problem in primitive variables, Numer. math., 33, 211-224, (1979) · Zbl 0423.65058 [9] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers, RAIRO anal. numér., R-2, 129-151, (1974) · Zbl 0338.90047 [10] Brezzi, F., A survey of mixed finite element methods, () · Zbl 0665.73058 [11] Brezzi, F.; Douglas, J., Stabilized mixed methods for the Stokes problem, Numer. math., (1986), (to appear) [12] Brezzi, F.; Douglas, J.; Marini, L.D., Two families of mixed finite elements for second order elliptic problems, Numer. math., 47, 217-233, (1985) · Zbl 0599.65072 [13] Brezzi, F.; Douglas, J.; Marini, L.D., Variable degree mixed methods for second order elliptic problems, Mat. apl. comput., 4, 1, 19-34, (1985) · Zbl 0592.65073 [14] Brezzi, F.; Pitkäranta, J., On the stabilization of finite element approximations of the Stokes equations, (), 11-19 · Zbl 0552.76002 [15] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/Petrov-Galerkin methods for advection dominated flows, () · Zbl 0449.76077 [16] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. meths. appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041 [17] Ciarlet, P.G., The finite element method for elliptic problems, (1978), North-Holland Amsterdam · Zbl 0445.73043 [18] Crouzeix, M.; Raviart, P.A., Conforming and non-conforming finite element methods for solving the stationary Stokes equations, RAIRO anal. numér., 7, R-3, 33-76, (1973) · Zbl 0302.65087 [19] Duvaut, G.; Lions, J.L., Inequalities in mechanics and physics, (1976), Springer Berlin · Zbl 0331.35002 [20] Fortin, M., An analysis of the convergence of mixed finite element methods, RAIRO anal. numér., 11, 4, 341-354, (1977) · Zbl 0373.65055 [21] Fortin, M., Old and new finite element methods for incompressible flows, Internat. J. numer. meths. fluids, 1, 347-364, (1981) · Zbl 0467.76030 [22] Franca, L.P., New mixed finite element methods, () · Zbl 0651.65078 [23] Franca, L.P.; Hughes, T.J.R.; Loula, A.F.D.; Miranda, I., A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element method, (), Also: Numer. Math. (to appear) [24] Girault, V.; Raviart, P.A., Finite element methods for Navier-Stokes equations, theory and algorithms, (1986), Springer Berlin · Zbl 0396.65070 [25] Hellinger, E., Der allgemeine ansatz der mechanik der kontinua, Encyclopädie math. wissensch., 4, 4, 602-694, (1914) [26] Hughes, T.J.R., Recent progress in the development and understanding of SUPG methods with special reference to the compressible Euler and Navier-Stokes equations, Internat. J. numer. methods fluids, (1987) · Zbl 0638.76080 [27] Hughes, T.J.R.; Brooks, A., A multidimensional upwind scheme with no crosswind diffusion, (), 19-35 · Zbl 0423.76067 [28] Hughes, T.J.R.; Brooks, A.N., A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: application to the streamline upwind procedure, (), 46-65 [29] Hughes, T.J.R.; Franca, L.P., A new finite element formulation for computational fluid dynamics: VII. the Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity/pressure spaces, Comput. meths. appl. mech. engrg., 65, 85-96, (1987) · Zbl 0635.76067 [30] Hughes, T.J.R.; Franca, L.P., A mixed finite element formulation for Reissner-Mindlin palte theory: uniform convergence of all higher-order spaces, Comput. meths. appl. mech. engrg., 67, 223-240, (1988) · Zbl 0611.73077 [31] Hughes, T.J.R.; Franca, L.P.; Balestra, M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuška-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accomodating equal-order interpolations, Comput. meths. appl. mech. engrg., 59, 85-99, (1986) · Zbl 0622.76077 [32] Johnson, C.; Nävert, U., An analysis of some finite element methods for advection-diffusion problems, (), 96-116 [33] Loula, A.F.D.; Franca, L.P.; Hughes, T.J.R.; Miranda, I., Stability, convergence and accuracy of a new finite element method for the circular arch problem, Comput. meths. appl. mech. engrg., 63, 281-303, (1987) · Zbl 0607.73077 [34] Loula, A.F.D.; Hughes, T.J.R.; Franca, L.P.; Miranda, I., Mixed Petrov-Galerkin method for the Timoshenko beam, Comput. meths. appl. mech. engrg., 63, 133-154, (1987) · Zbl 0607.73076 [35] Loula, A.F.D.; Miranda, I.; Hughes, T.J.R.; Franca, L.P., A successful mixed formulation for axisymmetric shell analysis employing discontinuous stress fields of the same order as the displacement field, (), 581-599 [36] Malkus, D.S.; Hughes, T.J.R., Mixed finite element methods—reduced and selective integration techniques: A unification of concepts, Comput. meths. appl. mech. engrg., 15, 63-81, (1978) · Zbl 0381.73075 [37] Nedelec, J.C., Mixed finite elements in R^{3}, Numer. math., 35, 315-341, (1980) · Zbl 0419.65069 [38] Oden, J.T.; Carey, G.F., () [39] Oden, J.T.; Jacquotte, O.P., Stability of some mixed finite element methods for Stokesian flows, Comput. meth. appl. mech. engrg., 43, 231-247, (1984) · Zbl 0598.76033 [40] Pitkäranta, J.; Stenberg, R., Analysis of some mixed finite element methods for plane elasticity equations, Math. comp., 41, 164, 399-423, (1983) · Zbl 0537.73057 [41] Raviart, P.A.; Thomas, J.M., A mixed finite element method for second order elliptic problems, (), 292-315 · Zbl 0362.65089 [42] Reissner, E., On a variational theorem in elasticity, J. math. phys., 29, 90-95, (1950) · Zbl 0039.40502 [43] Stenberg, R., Analysis of mixed finite element methods for the Stokes problem: A unified approach, Math. comp., 42, 9-23, (1984) · Zbl 0535.76037 [44] Stenberg, R., On the construction of optimal mixed finite element methods for the linear elasticity problem, Numer. math., 48, 447-462, (1986) · Zbl 0563.65072 [45] Taylor, C.; Hood, P., Numerical solution of the Navier-Stokes equations using the finite element technique, Computers & fluids, 1, 1-28, (1973) · Zbl 0328.76020 [46] Verfürth, R., Error estimates for a mixed finite element approximation of the Stokes equations, RAIRO anal. numér., 18, 175-182, (1984) · Zbl 0557.76037
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