A mixed finite element method for elliptic optimal control problems using a three-field formulation

In this paper, we consider an optimal control problem governed by elliptic differential equations posed in a three-field formulation. Using the gradient as a new unknown we write a weak equation for the gradient using a Lagrange multiplier. We use a biorthogonal system to discretise the gradient, which leads to a very efficient numerical scheme. A numerical example is presented to demonstrate the convergence of the finite element approach. References D. Boffi, F. Brezzi, and M. Fortin. Mixed finite element methods and applications. Springer–Verlag, 2013. doi:10.1007/978-3-642-36519-5. S.C. Brenner and L.R. Scott. The Mathematical Theory of Finite Element Methods. Springer–Verlag, New York, 1994. doi:10.1007/978-0-387-75934-0. Yanping Chen. Superconvergence of quadratic optimal control problems by triangular mixed finite element methods. International journal for numerical methods in engineering, 75(8):881–898, 2008. doi:10.1002/nme.2272. Hongfei Fu, Hongxing Rui, Jian Hou, and Haihong Li. A stabilized mixed finite element method for elliptic optimal control problems. Journal of Scientific Computing, 66(3):968–986, 2016. doi:10.1007/s10915-015-0050-3. Hui Guo, Hongfei Fu, and Jiansong Zhang. A splitting positive definite mixed finite element method for elliptic optimal control problem. Applied Mathematics and Computation, 219(24):11178–11190, August 2013. doi:10.1016/j.amc.2013.05.020. Muhammad Ilyas and Bishnu P. Lamichhane. A stabilised mixed finite element method for the poisson problem based on a three-field formulation. In M. Nelson, D. Mallet, B. Pincombe, and J. Bunder, editors, Proceedings of EMAC-2015, volume 57 of ANZIAM J., pages C177–C192. Cambridge University Press, 2016. doi:10.21914/anziamj.v57i0.10356. Bishnu P Lamichhane, AT McBride, and BD Reddy. A finite element method for a three-field formulation of linear elasticity based on biorthogonal systems. Computer Methods in Applied Mechanics and Engineering, 258:109–117, 2013. doi:10.1016/j.cma.2013.02.008. B.P. Lamichhane. Inf-sup stable finite element pairs based on dual meshes and bases for nearly incompressible elasticity. IMA Journal of Numerical Analysis, 29:404–420, 2009. doi:10.1093/imanum/drn013. B.P. Lamichhane. A mixed finite element method for the biharmonic problem using biorthogonal or quasi-biorthogonal systems. Journal of Scientific Computing, 46:379–396, 2011. doi:10.1007/s10915-010-9409-7. B.P. Lamichhane and E. Stephan. A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. Numerical Methods for Partial Differential Equations, 28:1336–1353, 2012. doi:10.1002/num.20683. Xianbing Luo, Yanping Chen, and Yunqing Huang. Some error estimates of finite volume element approximation for elliptic optimal control problems. International Journal of Numerical Analysis and Modeling, 10(3):697–711, 2013. http://www.math.ualberta.ca/ijnam/Volume-10-2013/No-3-13/2013-03-11.pdf. Fredi Troltzsch. On finite element error estimates for optimal control problems with elliptic PDEs. In International Conference on Large-Scale Scientific Computing, pages 40–53. Springer, 2009. doi:10.1007/978-3-642-$12535-5_4$. Fredi Troltzsch. Optimal control of partial differential equations, volume 112. American Mathematical Society, 2010. http://www.ams.org/books/gsm/112/

Mortar Finite Elemente höherer Ordnung mit dualen Lagrange-Multiplikatorräumen und Anwendungen

The numerical approximation of partial differential equations coming from physical and engineering modeling is often a challenging task. Most often these partial differential equations are discretized with finite elements and can be solved by modern super-computers. Working with different discretization techniques in different subdomains or independent triangulations, the challenging task is to couple these different discretization schemes or non-matching triangulation without losing the optimality of the approach. Mortar methods yield optimal and flexible coupling techniques for different discretization schemes. Especially when combined with dual Lagrange multiplier spaces, the efficient realization of the weak matching condition is possible, and efficient multigrid methods can be adapted to the non-conforming situation. In this thesis, we concentrate on higher order dual Lagrange multiplier spaces for mortar finite elements. These non-standard Lagrange multipliers show the same qualitative a priori estimates and quantitative numerical results as the standard ones and yield locally supported basis functions for the constrained space leading to an efficient numerical realization. Working with abstract assumptions on Lagrange multiplier spaces, we prove optimal a priori estimates for mortar finite elements allowing that the dimension of the Lagrange multiplier space can be smaller than the dimension of the trace space of the finite element space from the slave side (with zero boundary condition on the interface). Geometrically non-conforming decompositions and locally refined meshes are also covered. In two dimensions, we show that a dual Lagrange multiplier space can be constructed for a finite element space of any order satisfying these abstract assumptions. In contrast to earlier approaches, these Lagrange multiplier basis functions have the same support as the nodal finite element basis functions. Using an interesting relation between biorthogonality and quadrature formulas, we prove that an optimal dual Lagrange multiplier space for a finite element space can be constructed if and only if the finite element space is based on Gau-Lobatto nodes. The two-dimensional construction can easily be extended to the three-dimensional case for a finite element space with tensor product structure. If a finite element space does not have the tensor product structure, e.g., serendipity elements on hexahedra or conforming finite elements on simplices, the situation is more difficult. To deal with this problem, we generalize the idea of a dual Lagrange multiplier space by introducing a quasi-dual Lagrange multiplier space for quadratic serendipity elements. Furthermore, working with a more general assumption that the Lagrange multiplier space can have smaller dimension than the trace space of the finite element space at the slave side (with zero boundary condition on the interface), we introduce dual Lagrange multiplier spaces for quadratic tetrahedral and serendipity elements. Numerical results are presented to illustrate the performance of our approach. We also study interface problems arising from heat conduction with a discontinuous flux and solution within the framework of mortar finite element methods. Taking into account non-homogeneous jumps of the solution and the flux across the interface, we give a saddle point formulation of the interface problems. Optimal a priori estimates are proved and numerical results are provided. We have applied mortar finite elements to couple different physical models, material laws and discretization schemes. Furthermore, time-dependent heat transfer problems with sliding meshes are also considered. Another particular interest for us is the locking phenomenon in linear and nonlinear elasticity. We analyze low order finite element methods based on the Hu-Washizu formulation in linear elasticity and prove the robust and optimal convergence of the finite element approximation of the displacement for the nearly incompressible case. A three-field mixed formulation for finite elasticity is also introduced and numerical results are presented. Mortar finite elements for coupling two different materials in elasticity, where one is a nearly incompressible material and the other one is a compressible material are analyzed for the linear elastic case, and numerical results are provided to verify the theoretical results.In dieser Arbeit stehen duale Lagrange-Multiplikatorräume höherer Ordnung für Mortar-Finite-Elemente im Mittelpunkt. Diese speziellen Lagrange-Multiplikatoren weisen die gleichen qualitativen und quantitativen numerischen Eigenschaften auf wie Standard-Lagrange-Multiplikatoren und liefern Basisfunktionen mit lokalem Träger für den die Kopplungsbedingung respektierenden Raum, was zu einer effizienten numerischen Unsetzung führt. Indem mit abstrakten Annahmen für die Lagrange-Multiplikatorräume gearbeitet wird, können a priori Abschätzungen für Mortar-Finite-Elemente gezeigt werden, die es erlauben, dass die Dimension des Lagrange-Multiplikatorraumes kleiner sein darf als die Dimension der Spur des finite Element-Raumes, der die Null-Randbedingung am Interface der Slave-Seite erfüllt. Geometrisch nicht-konforme Zerlegungen und lokal verfeinerte Gitter sind auch verwendbar. Für zweidimensionale Mortar-Finite-Elemente wird gezeigt, dass ein dualer Lagrange-Multiplikatorraum für einen Finite-Element-Raum beliebiger Ordnung konstruiert werden kann, der diesen abstrakten Annahmen genügt. Im Gegensatz zu früheren Ansätzen haben diese Lagrange-Multiplikator-Basisfunktionen den gleichen Träger wie die nodalen Finiten-Element-Basisfunktionen. Indem eine interessante Beziehung zwischen der Biorthogonalität und Quadraturformeln verwendet wird, wird bewiesen, dass ein optimaler dualer Lagrange-Multiplikatorraum für einen Finite-Element-Raum nur dann konstruiert werden kann, wenn der Finite-Element-Raum auf Gau-Lobatto-Knoten basiert. Das zweidimensionale Konstruktionsschema kann leicht auf den dreidimensionalen Fall erweitert werden, sofern ein Finite-Element-Raum mit Tensorprodukt-Struktur vorliegt. Wenn ein Finite-Element-Raum keine Tensorprodukt-Struktur aufweist, wie z.B. bei Serendipity-Elementen auf Hexaedernetz oder bei konforme simplizialen Elementen, ist die Lage schwieriger. Um dieses Problem zu behandeln, wird die Idee des dualen Lagrange-Multiplikatorraums verallgemeinert, indem ein quasi-dualer Lagrange-Multiplikatorraum für quadratische Serendipity-Elemente eingeführt wird. Ferner werden anhand der allgemeineren Annahme, dass der Lagrange-Multiplikatorraum eine kleinere Dimension als der Spur-Raum des Approximationsraums auf der Slave-Seite (mit Null-Randbedingung am Interface) haben kann, duale Lagrange-Multiplikatorräume für quadratische simpliziale- und Serendipity-Elemente eingeführt. Numerische Ergebnisse demonstrieren die Effizienz des Ansatzes. Des Weiteren werden Interface-Probleme im Rahmen der Mortar-Finite-Element-Methoden behandelt, die aus der Wärmeleitung mit unstetigem Fluss herrühren. Unter Berücksichtigung von inhomogenen Sprüngen in der Lösung und im Fluss am Interface wird eine Sattelpunktformulierung des Interface-Problemes hergeleitet. Deren Optimalität wird bewiesen und durch numerische Ergebnisse untermauert. Zusätzlich werden Mortar-Finite-Elemente zur Kopplung verschiedener physikalischer Modelle, Materialgesetze und Diskretisierungs-Schemata angewandt. Der letzte wichtige Punkt, der in dieser Arbeit beleuchtet wird, ist das sogenannte 'Locking'-Phänomen in der linearen und nichtlinearen Elastizität. Analysiert werden Finite-Element-Methoden von niedrigster Ordnung, die auf der Hu-Washizu-Formulierung basieren. Es wird gezeigt, dass die numerische Approximation robust und optimal konvergiert. Eine Drei-Feld-gemischte Formulierung für nichtlineare (finite) Elastizität wird ebenfalls eingeführt und numerische Ergebnisse dazu präsentiert. Der Fall finiter Elemente zur Kopplung zweier verschiedener Materialien, wobei eines davon nahezu inkompressibel und das andere kompressibel ist, wird für den linear-elastischen Fall analysiert; numerische Ergebnisse bestätigen die theoretischen Aussagen

A mixed finite element method based on a biorthogonal system for nearly incompressible elastic problems

A Petrov--Galerkin scheme in a saddle point formulation gives rise to a non-symmetric saddle point problem. This article considers a non-symmetric saddle point problem with a penalty parameter. A mixed finite element method for linear elasticity based on a Petrov--Galerkin formulation is then analyzed within the framework of the non-symmetric saddle point problem with penalty. Working with a biorthogonal system to discretize the pressure equation, we obtain a robust and efficient numerical scheme for nearly incompressible linear elasticity using linear finite elements. A numerical example demonstrates the robustness of the approach. These results are useful to analyze a Petrov--Galerkin scheme in a saddle point problem. References Babuska, I. and Suri, M. (1992). Locking effects in the finite element approximation of elasticity problems. Numerische Mathematik, 62, 439--463. doi:10.1007/BF01396238 Babuska, I. and Suri, M. (1992). On locking and robustness in the finite element method. SIAM Journal on Numerical Analysis, 29, 1261--1293. doi:10.1137/0729075 Brezzi, F. and Fortin, M. (1991). Mixed and hybrid finite element methods. Springer--Verlag, New York. Braess, D. (1996). Stability of saddle point problems with penalty. Mathematical Modelling and Numerical Analysis, 30, 731--742. Lamichhane, B. (2008). Inf-sup stable finite element pairs based on dual meshes and bases for nearly incompressible elasticity. IMA Journal of Numerical Analysis. doi:10.1093/imanum/drn013 Nicolaides, R. (1982). Existence, uniqueness and approximation for generalized saddle point problems. SIAM Journal on Numerical Analysis, 19, 349--357. doi:10.1137/0719021 Bernardi, C., Canuto, C. and Maday, Y. (1988). Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem. SIAM Journal on Numerical Analysis, 25, 1237--1271. doi:10.1137/0725070 Ciarlet, P. J., Huang, J. and Zou, J. (2003). Some observations on generalized saddle-point problems. SIAM Journal on Matrix Analysis and Applications, 25, 224--236. doi:10.1137/S0895479802410827 Arnold, D. N., Brezzi, F. and Fortin, M. (1984). A stable finite element for the Stokes equations. Calcolo, 21, 337--344. doi:10.1007/BF02576171 Braess, D. (2001). Finite Elements. Theory, fast solver, and applications in solid mechanics. Cambridge University Press, Second Edition. Brenner, S. (1993). A nonconforming mixed multigrid method for the pure displacement problem in planar linear elasticity. SIAM Journal on Numerical Analysis, 30, 116--135. doi:10.1137/073000

Removing a mixture of Gaussian and impulsive noise using the total variation functional and split Bregman iterative method

We apply the split Bregman iterative method to minimise the total variation of a piecewise polynomial function to remove Gaussian and impulsive noise from an image. We compare these numerical results with another approach based on the gradient penalty. Both approaches use a finite element method. Numerical results show that the method based on the total variation functional is superior only for one class of images. References Plataniotis, K. N. and Venetsanopoulos, A. N. Color Image Processing and Applications. Springer, 2000. doi:10.1007/978-3-662-04186-4 Chan, T. F. and Shen, J. Image Processing And Analysis: Variational, PDE, Wavelet, And Stochastic Methods. SIAM, 2005. doi:10.1137/1.9780898717877 Blanchet, G. and Charbit, M. Digital Signal and Image Processing using Matlab. Wiley, 2010. doi:10.1002/9780470612385. Preusser, T. and Rumpf, M. An adaptive finite element method for large scale image processing. J. Vis. Commun. Image R. 11:183–195 2000. doi:10.1006/jvci.1999.0444 Ferrant, M., Warfield, S. K., Nabavi, A., Jolesz, F. A. and Kikinis, R. Registration of 3D intraoperative MR images of the brain using a finite-element biomechanical model. IEEE T. Med. Imaging 20:1384–1397, 2001. doi:10.1007/978-3-540-40899-4_3 Besdok, E. Impulsive noise suppression from images by using Anfis interpolant and Lillietest. EURASIP J. Appl. Sig. P. 2004:526574, 2004. doi:10.1155/S1110865704403126 Wang, Z., Qi, F. and Zhou, F. A discontinuous finite element method for image denoising. In Image Analysis and Recognition vol. 4141 of Lecture Notes in Computer Science. Springer Berlin/Heidelberg, 2006. doi:10.1007/11867586_11 Demaret, L. and Iske, A. Adaptive image approximation by linear splines over locally optimal Delaunay triangulations. IEEE Signal Proc. Lett. 13:281–284, 2006. doi:10.1109/LSP.2006.870358 Aizenberg, I. and Butakoff, C. Effective impulse detector based on rank-order criteria. IEEE Signal Proc. Lett. 11:363–366, 2004. doi:10.1109/LSP.2003.822925. Garnett, R., Huegerich, T., Chui, C. and He, W. A universal noise removal algorithm with an impulse detector. IEEE T. Image Process. 14:1747–1754, 2005. doi:10.1109/TIP.2005.857261 Donoho, D. L. De-noising by soft-thresholding. IEEE T. Inform. Theory 41:613–627, 1995. doi:10.1109/18.382009. Donoho, D. L. and Johnstone, J. M. Ideal spatial adaptation by wavelet shrinkage. Biometrika 81:425–455, 1994. doi: 10.1093/biomet/81.3.425 Luisier, F., Blu, T. and Unser, M. A new SURE approach to image denoising: interscale orthonormal wavelet thresholding. IEEE T. Image Process. 16:593–606, 2007. doi:10.1109/TIP.2007.891064 Xu, Q., Ma, L., Li, M., Wang, W., Cai, J., Brunelli, R. and Messelodi, S. Fuzzy weighted average filtering for mixture noises. In Third International Conference on Image and Graphics pp. 18–21, 2004. doi:10.1109/ICIG.2004.70 Lamichhane, B. P. Finite element techniques for removing mixture of gaussian and impulsive noise. IEEE T. Signal Process. 57:2538–2547, 2009. doi:10.1109/TSP.2009.2016272 Duchon, J. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics 571:85–100. Springer-Verlag Berlin, 1977. doi:10.1007/BFb0086566 Wahba, G. Spline Models for Observational Data, vol. 59, of Series in Applied Mathematic. SIAM, Philadelphia, 1990. doi:10.1137/1.9781611970128 Quarteroni, A. and Valli, A. Numerical approximation of partial differential equations. Springer–Verlag, Berlin, 1994. doi:10.1007/978-3-540-85268-1 Goldstein, T. and Osher, S. The split Bregman method for L1-regularized problems. SIAM J. Imaging Sci. 2:323–343, 2009. doi:10.1137/080725891 Wang, Y., Yang, J., Yin, W. and Zhang, Y. A new alternating minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1:248–272, 2008. doi:10.1137/080724265 van den Doel, K., Ascher, U. M. and Haber, E. The lost honour of l2-based regularization. In Large Scale Inverse Problems Radon Series on Computational and Applied Mathematics 13:181–203. De Gruyter, 2012. http://www.degruyter.com/viewbooktoc/product/18202

A new minimization principle for the Poisson equation leading to a flexible finite element approach

A new minimization principle for the Poisson equation using two variables – the solution and the gradient of the solution – is introduced. This principle allows us to use any conforming finite element spaces for both variables, where the finite element spaces do not need to satisfy the so-called inf–sup condition. A numerical example demonstrates the superiority of this approach. doi:10.1017/S144618111700030

A mixed finite element method using a biorthogonal system for optimal control problems governed by a biharmonic equation

In this article, we consider an optimal control problem governed by a biharmonic equation with clamped boundary conditions. We use the Ciarlet--Raviart formulation combined with a biorthogonal system to obtain an efficient numerical scheme. We discuss the a priori error analysis and present results of the numerical experiments that validate the theoretical estimates. References L. Boudjaj, A. Naji, and F. Ghafrani. Solving biharmonic equation as an optimal control problem using localized radial basis functions collocation method. Eng. Anal. Bound. Elements 107 (2019), pp. 208–217. doi: 10.1016/j.enganabound.2019.07.007 W. Cao and D. Yang. Ciarlet–Raviart mixed finite element approximation for an optimal control problem governed by the first biharmonic equation. J. Comput. App. Math. 233.2 (2009), pp. 372–388. doi: 10.1016/j.cam.2009.07.039 P. G. Ciarlet. The finite element method for elliptic problems. Vol. 40. Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2002. doi: 10.1137/1.9780898719208. V. Girault and P.-A. Raviart. Finite element methods for Navier–Stokes equations. Vol. 5. Springer Series in Computational Mathematics. Springer-Verlag, 1986. doi: 10.1007/978-3-642-61623-5 T. Gudi, N. Nataraj, and K. Porwal. An interior penalty method for distributed optimal control problems governed by the biharmonic operator. Comput. Math. App. 68.12 (2014), pp. 2205–2221. doi: 10.1016/j.camwa.2014.08.012 B. P. Lamichhane. A mixed finite element method for the biharmonic problem using biorthogonal or quasi-biorthogonal systems. J. Sci. Comput. 46.3 (2011), pp. 379–396. doi: 10.1007/s10915-010-9409-7. B. P. Lamichhane and E. Stephan. A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. Numer. Meth. Part. Diff. Eq. 28 (2012), pp. 1336–1353. doi: 10.1002/num.20683 J. L. Lions. Optimal control of systems governed by partial differential equations. Vol. 170. Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Berlin, 1971. url: https://link.springer.com/book/9783642650260 F. Tröltzsch. Optimal control of partial differential equations: Theory, methods and applications. Vol. 112. Graduate Studies in Mathematics. American Mathematical Society, 2010. doi: 10.1090/gsm/112. G. N. Wells, E. Kuhl, and K. Garikipati. A discontinuous Galerkin method for the Cahn–Hilliard equation. J. Comput. Phys. 218 (2006), pp. 860 –877. doi: 10.1016/j.jcp.2006.03.01

A stabilised mixed finite element method for thin plate splines based on biorthogonal systems

We propose a novel stabilised mixed finite element method for the discretisation of thin plate splines. The mixed formulation is obtained by introducing the gradient of the smoother as an additional unknown. Working with a pair of bases for the gradient of the smoother and the Lagrange multiplier, which forms a biorthogonal system, we eliminate these two variables (gradient of the smoother and Lagrange multiplier) leading to a positive definite formulation. We prove a sub-optimal a priori error estimate for the proposed finite element scheme. References A. Bab-Hadiashar, D. Suter, and R. Jarvis. Optic flow computation using interpolating thin-plate splines. In Second Asian Conference on Computer Vision (ACCV'95), pages 452--456, Singapore, 1995. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.2380. C. Bernardi, Y. Maday, and A. T. Patera. A new nonconforming approach to domain decomposition: the mortar element method. In H. Brezzi et al., editor, Nonlinear partial differential equations and their applications, pages 13--51. Pitman, 1994. J. Brandts and M. Kr\T1\i zek. Gradient superconvergence on uniform simplicial partitions of polytopes. IMA Journal of Numerical Analysis, 23:489--505, 2003. doi:10.1093/imanum/23.3.489. S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer--Verlag, New York, 1994. X. Cheng, W. Han, and H. Huang. Some mixed finite element methods for biharmonic equation. Journal of Computational and Applied Mathematics, 126:91--109, 2000. doi:10.1016/S0377-0427(99)00342-8. P. G Ciarlet. The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. J. Duchon. Splines minimizing rotation-invariant semi-norms in Sobolev spaces. In Constructive Theory of Functions of Several Variables, Lecture Notes in Mathematics, volume 571, pages 85--100. Springer-Verlag, Berlin, 1977. A. Iske. Multiresolution Methods in Scattered Data Modelling, volume 37 of LNCS. Springer, Heidelberg, 2004. C. Johnson and J. Pitkaranta. Some mixed finite element methods related to reduced integration. Mathematics of Computation, 38:375--400, 1982. doi:10.1090/S0025-5718-1982-0645657-2. C. Kim, R. D. Lazarov, J. E. Pasciak, and P. S. Vassilevski. Multiplier spaces for the mortar finite element method in three dimensions. SIAM Journal on Numerical Analysis, 39:519--538, 2001. doi:10.1137/S0036142900367065. B. P. Lamichhane. Higher Order Mortar Finite Elements with Dual Lagrange Multiplier Spaces and Applications. LAP LAMBERT Academic Publishing, 2011. B. P. Lamichhane. A stabilized mixed finite element method for the biharmonic equation based on biorthogonal systems. Journal of Computational and Applied Mathematics, 235:5188--5197, 2011. doi:10.1016/j.cam.2011.05.005. B. P. Lamichhane, S. Roberts, and L. Stals. A mixed finite element discretisation of thin-plate splines. In W. McLean and A. J. Roberts, editors, Proceedings of the 15th Biennial Computational Techniques and Applications Conference, CTAC-2010, volume 52 of ANZIAM J., pages C518--C534, 2011. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/3934. S. Roberts, M. Hegland, and I. Altas. Approximation of a thin plate spline smoother using continuous piecewise polynomial functions. SIAM Journal on Numerical Analysis, 41:208--234, 2003. doi:10.1137/S0036142901383296. D. B. Szyld. The many proofs of an identity on the norm of oblique projections. Numerical Algorithms, 42:309--323, 2006. http://link.springer.com/article/10.1007%2Fs11075-006-9046-2. G. Wahba. Spline Models for Observational Data, volume 59 of Series in Applied Mathematic. SIAM, Philadelphia, first edition, 1990. H. Wendland. Scattered Data Approximation. Cambridge University Press, first edition, 2005

A stabilized mixed finite element method for Poisson problem based on a three-field formulation

We present a mixed finite element method for a three-field formulation of the Poisson problem and apply a biorthogonal system leading to an efficient numerical computation. The three-field formulation is similar to the Hu-Washizu formulation for the linear elasticity problem. A parameterised approach is given to stabilise the problem so that its associated bilinear form is coercive on the whole space. Analysis of optimal choices of parameter approximation and numerical examples are provided to evaluate our stabilised form. References P. B. Bochev and C. R. Dohrmann. A computational study of stabilized, low-order $C^0$ finite element approximations of Darcy equations. Comput. Mech. 38(4):323–333, 2006. doi:10.1007/s00466-006-0036-y D. Braess. Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, 3rd edition, 2007. doi:10.1017/CBO9780511618635 S. C. Brenner and L. R. Scott. The Mathematical Theory of Finite Element Methods. Springer, New York, 3rd edition edition, 2008. doi:10.1007/978-0-387-75934-0 B. P. Lamichhane. Two simple finite element methods for Reissner–Mindlin plates with clamped boundary condition. Appl. Numer. Math. 72:91–98, 2013. doi:10.1016/j.apnum.2013.04.005 B. P. Lamichhane, A. T. McBride, and B. D. Reddy. A finite element method for a three-field formulation of linear elasticity based on biorthogonal systems. Comput. Method. Appl. Mech. Eng. 258:109–117, 2013. doi:10.1016/j.cma.2013.02.008 B. P. Lamichhane and E. P. Stephan. A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems. Numer. Meth. Part. D. E. 28(4):1336–1353, 2012. doi:10.1002/num.20683 S. Micheletti and R. Sacco. Dual-primal mixed finite elements for elliptic problems. Comput. Method. Appl. Mech. Eng. 17(2):137–151, 2001. doi:10.1002/1098-2426(200103)17:2<137::AID-NUM4>3.0.CO;2-0 W. F. Mitchell. A collection of 2D elliptic problems for testing adaptive grid refinement algorithms. Appl. Math. Comput. 220:350–364, 2013. doi:10.1016/j.amc.2013.05.06

A new MITC finite element method for Reissner--Mindlin plate problem based on a biorthogonal system

We present a new MITC (Mixed Interpolated Tensorial Components) finite element method for Reissner--Mindlin plate equations. The new finite element method uses a biorthogonal system to construct the reduction operator for the MITC element. Numerical results are shown to demonstrate the performance of the approach. References D. Arnold and F. Brezzi. Some new elements for the Reissner–Mindlin plate model. In J. L. Lions, C. Baiocchi and E. Magenes editors. Boundary Value Problems for Partial Differential Equations and Applications: Dedicated to E. Magenes. Masson, Paris, 1993, pp. 287–292. http://umn.edu/ arnold/papers/rmelts.pdf D. Arnold, F. Brezzi and M. Fortin. A stable finite element for the Stokes equations. Calcolo, 21:337–344, 1984. doi:10.1007/BF02576171 D. Arnold and R. Falk. A uniformly accurate finite element method for the Reissner–Mindlin plate. SIAM Journal on Numerical Analysis, 26:1276–1290, 1989. doi:10.1137/0726074 D. Arnold and R. Falk. Analysis of a linear-linear finite element for the Reissner–Mindlin plate model. Mathematical Models and Methods in Applied Science, 7:217–238, 1997. doi:10.1142/S0218202597000141 D. Boffi, F. Brezzi and M. Fortin. Mixed Finite Element Methods and Applications. Springer–Verlag, Berlin, Heidelberg, 2013. doi:10.1007/978-3-642-36519-5 D. Braess. Finite Elements. Theory, Fast Solver, and Applications in Solid Mechanics. 2nd edition, Cambridge Univ. Press, Cambridge, 2001. doi:10.1017/CBO9780511618635 S. Brenner. Multigrid methods for parameter dependent problems. ESAIM: Mathematical Modelling and Numerical Analysis, 30:265–297, 1996. doi:10.1051/m2an/1996300302651 S. Brenner and L. Scott. The Mathematical Theory of Finite Element Methods. Springer–Verlag, New York, 1994. doi:10.1007/978-0-387-75934-0 C. Chinosi, C. Lovadina and L. Marini. Nonconforming locking-free finite elements for Reissner–Mindlin plates. Computer Methods in Applied Mechanics and Engineering, 195:3448–3460, 2006. doi:10.1016/j.cma.2005.06.025 L. B. da Veiga, C. Chinosi, C. Lovadina and L. F. Pavarino. Robust BDDC preconditioners for Reissner–Mindlin plate bending problems and MITC elements. SIAM Journal on Numerical Analysis, 47(6):4214–4238, 2010. doi:10.1137/080717729 E. Dvorkin and K. Bathe. A continuum mechanics based four-node shell element for general nonlinear analysis. Engineering Computations, 1:77–88, 1984. doi:10.1108/eb023562 R. Falk and T. Tu. Locking-free finite elements for the Reissner-Mindlin plate. Mathematics of Computation, 69:911–928. doi:10.1090/S0025-5718-99-01165-5 C. Kim, R. Lazarov, J. Pasciak and P. Vassilevski. Multiplier spaces for the mortar finite element method in three dimensions. SIAM Journal on Numerical Analysis, 39:519–538, 2001. doi:10.1137/S0036142900367065 B. Lamichhane. Higher Order Mortar Finite Elements with Dual Lagrange Multiplier Spaces and Applications. PhD thesis, University of Stuttgart, 2006. doi:10.18419/opus-4770 B. Lamichhane. Two simple finite element methods for Reissner–Mindlin plates with clamped boundary condition. Applied Numerical Mathematics, 72:91–98, 2013. doi:10.1016/j.apnum.2013.04.005 C. Lovadina. A new class of mixed finite element methods for Reissner-Mindlin plates. SIAM Journal on Numerical Analysis, 33:2456–2467, 1996. doi:10.1137/S0036142994265061 C. Lovadina. A low-order nonconforming finite element for Reissner–Mindlin plates. SIAM Journal on Numerical Analysis, 42:2688–2705, 2005. doi:10.1137/040603474 M. Lyly, J. Niiranen and R. Stenberg. A refined error analysis of MITC plate elements. Mathematical Models and Methods in Applied Sciences, 16:967–977, 2006. doi:10.1142/S021820250600142X D. Mijuca. On hexahedral finite element HC8/27 in elasticity. Computational Mechanics, 33:466–480, 2004. doi:10.1007/s00466-003-0546-9 S. Timoshenko and J. Goodier. Theory of Elasticity. 3rd edition, McGraw-Hill, New York, 1970. doi:10.1017/S036839310012471