A General Purpose Variational Formulation for Boundary Value Problems of Orders Greater than Two

Document Type : Research Paper


Department of Mathematics and Computer Science, Loyola University, New Orleans, LA 70118, USA


We develop a new general purpose variational formulation, particularly suitable for solving boundary value problems of orders greater than two. The functional related to this variational formulation requires only Η1 regularity in order to be well-defined. Using the finite element method based on this new formulation thus becomes simple even for domains in dimensions greater than one.  We prove that a saddle-point solution to the new variational formulation is a weak solution to the associated boundary value problem. We also prove the convergence of the numerical methods used to find approximate solutions to the new formulation. We provide numerical tests to demonstrate the efficacy of this new paradigm.


Main Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Ciarlet, P.G., The Finite Element Method for Elliptic Problems, Elsevier North-Holland, Inc., 1st ed., 1978.
[2] Axelsson, O., Barker, V.A., Finite Element Solution of Boundary Value Problems. Theory and Computation, SIAM, 2001.
[3] Oden, J.T., Generalized conjugate functions for mixed finite element approximations of boundary value problems, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, 1972, 629–669.
[4] Babuska, I., Finite element method with lagrangian multipliers, Numer. Math., 1973, 20, 179–192.
[5] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 1974, R-2, 129–151.
[6] Brezzi, F., Raviart, P.A., Mixed finite element methods for 4th order elliptic equations, Topics in numerical analysis III: proceedings of the Royal Irish Academy Conference on Numerical Analysis, 1976, 33–56.
[7] Raviart, P.A., Thomas, J.M., A mixed finite element method for 2nd order elliptic problems, Mathematical aspects of finite element methods: proceedings of the conference held in Rome, 10-12 December 1975, 292–315.
[8] Falk, R.S., Approximation of the biharmonic equation by a mixed finite element method, SIAM J. Numer. Anal., 1978, 15(2), 556–567.
[9] Han, H.D., Nonconfirming elements in the mixed finite element method, Journal of Computational Mathematics, 1984, 2(3), 223–233.
[10] Monk, P., A mixed finite element method for the biharmonic equation, SIAM J. NUMER. ANAL., 1987, 24(4), 737–749.
[11] Brenner, S.C., A multigrid algorithm for the lowest-order raviart-thomas mixed triangular finite element method, SIAM J. NUMER. ANAL., 1992, 29(3), 647–678.
[12] Figueroa, L.E., Gatica, G.N., Márquez, A., Augmented mixed finite element methods for the stationary stokes equations, SIAM J. SCI. COMPUT., 2008, 31(2), 1082–1119.
[13] Camaño, J., Gatica, G.N., Oyarzúa, R., Tierra, G., An augmented mixed finite element method for the navier-stokes equations with variable viscosity, SIAM J. NUMER. ANAL., 2016, 54(2), 1069–1092.
[14] Barnafi, N., Gatica, G.N., Hurtado, D.E., Primal and mixed finite element methods for deformable image registration problems, SIAM Journal on Imaging Sciences, 2018, 11(4), 2529–2567.
[15] Lee, J.J., Piersanti, E., Mardal, K.A., Rognes, M.E., A mixed finite element method for nearly incompressible multiple-network poroelasticity, SIAM Journal on Scientific Computing, 2019, 41(2), A722–A747.
[16] Ambartsumyan, I., Khattatov, E., Nordbotten, J.M., Yotov, I., A multipoint stress mixed finite element method for elasticity on simplicial grids, SIAM Journal on Numerical Analysis, 2020, 58(1), 630–656.
[17] Carstensen, C., Ma, R., Adaptive mixed finite element methods for non-self-adjoint indefinite second-order elliptic pdes with optimal rates, SIAM Journal on Numerical Analysis, 2021, 59(2), 955–982.
[18] He, J.H., Modified lagrange multiplier method and generalized variational principle in fluid mechanics, JOURNAL OF SHANGHAI UNIVERSITY, 1997, 1(2), 117–122.
[19] He, J.H., Semi-inverse method and generalized variational principles with multi-variables in elasticity, Applied Mathematics and Mechanics, 2000, 21(7), 797–808.
[20] He, J.H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons and Fractals, 2004, 19, 847–851.
[21] He, J.H., Anjum, N., Skrzypacz, P.S., A variational principle for a nonlinear oscillator arising in the microelectromechanical system, Journal of Applied and Computational Mechanics, 2021, 7(1), 78–83.
[22] He, J.H., An alternative approach to establishment of a variational principle for the torsional problem of piezoelastic beams, Applied Mathematics Letters, 2016, 52, 1–3.
[23] He, J.H., Lagrange crisis and generalized variational principle for 3d unsteady flow, International Journal of Numerical Methods for Heat & Fluid Flow, 2020, 30(3), 1189–1196.
[24] EL-Kalaawy, O.H., New variational principle–exact solutions and conservation laws for modified ion-acoustic shock waves and double layers with electron degenerate in plasma, Physics of Plasmas, 2017, 24, 032308.
[25] He, J.H., A modified li-he’s variational principle for plasma, International Journal of Numerical Methods for Heat & Fluid Flow, 2021, 31(5), 1369–1372.
[26] He, J.H., Variational principle for the generalized kdv-burgers equation with fractal derivatives for shallow water waves, Journal of Applied and Computational Mechanics, 2020, 6(4), 735–740.
[27] Cao, X.Q., Guo, Y.N., Zhang, C.Z., Hou, S.C., Peng, K.C., Different groups of variational principles for whitham-broer-kaup equations in shallow water, Journal of Applied and Computational Mechanics, 2020, 6(Special Issue), 1178–1183.
[28] Cao, X.Q., Peng, K.C., Liu, M.Z., Zhang, C.Z., Guo, Y.N., Variational principles for two compound nonlinear equations with variable coefficients, Journal of Applied and Computational Mechanics, 2021, 7(2), 415–421.
[29] Liu, M.Z., Cao, X.Q., Zhu, X.Q., Liu, B.N., Peng, K.C., Variational principles and solitary wave solutions of generalized nonlinear schrödinger equation in the ocean, Journal of Applied and Computational Mechanics, 2021, doi: 10.22055/JACM.2021.36690.2890.
[30] Fortin, M., Glowinski, R., Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, NORTH-HOLLAND, 1st ed., 1983.
[31] Chien, W.Z., Method of high-order lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals, Applied Mcahematies and Mechanics, 1983, 4(2), 137–150.
[32] Wei, D., Li, X., Finite element solutions of cantilever and fixed actuator beams using augmented lagrangian methods, Journal of Applied and Computational Mechanics, 2018, 4, 125—132.
[33] Adomian, G., A review of the decomposition method in applied mathematics, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 1988, 135, 501–544.
[34] Adomian, G., A review of the decomposition method and some recent results for nonlinear equations, Computers & Mathematics with Applications, 1991, 21(5), 101–127.
[35] Zeidan, D., Chau, C.K., Lu, T.T., On the characteristic adomian decomposition method for the riemann problem, Math Meth Appl Sci., 2019, special issue, 1–16.
[36] Sil, S., Sekhar, T.R., Zeidan, D., Nonlocal conservation laws, nonlocal symmetries and exact solutions of an integrable soliton equation, Chaos, Solitons and Fractals, 2020, 139, 1–9.
[37] Zeidan, D., Chau, C.K., Lu, T.T., Zheng, W.Q., Mathematical studies of the solution of burgers’ equations by adomian decomposition method, Math Meth Appl Sci., 2020, 43, 2171–2188.