Seminar Numerical Analysis: Virtual Element Methods

Date	Speaker	Supervisor	Title	Literature
2018-10-01	U. Langer		Introduction
2018-10-16	L. Mitter	U. Langer	Polygonal and polyhedral meshes	[1], Section 2.3, pp. 16 - 30
2018-10-23	B. Endtmayer	U. Langer	Basis and interpolation operators (smooth functions)	[1], Section 2.3 - 2.5, pp. 31 - 51
2018-10-30	A. Schafelner	U. Langer	Interpolation of non-smooth functions	[1], Section 3.1 - 3.3, pp. 59 - 70
2018-11-13	R. Schneckenleitner	U. Langer	Anisotropic Case	[1], Section 3.4, pp. 70 - 90
2018-11-20	D. Jodlbauer	W. Zulehner	Basic Principles of Virtual Element Methods	[2]
2018-12-04	D. Jodlbauer	W. Zulehner	Computable VE approximation	[3]
2019-01-08	R. Schneckenleitner	U. Langer	VEM for general 2nd-order elliptic BVP	[4]
2019-01-15	B. Endtmayer	U. Langer	A posteriori error estimates for the VEM and AVEM	[5]
2019-01-22	A. Schafelner	W. Zulehner	Divergence free VEM for Stokes	[6]
2019-01-29	L. Mitter	W. Zulehner	VEM for plates	[7]

Literature

[1] S. Weißer, BEM-based Finite Element Approaches on Polytopal Meshes, Manuskript, 2018.
[2] L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L.D. Marini, A. Russo, Basicprinciples of virtual element methods, Math. Models Methods Appl. Sci. 23 (1), 199–214 (2013).
[3] L. Beirao da Veiga, F. Brezzi, L.D. Marini, A. Russo, The hitchhiker's guide to the virtual element method, Math. Models Methods Appl. Sci. 24, 1541–1573 (2014).
[4] L. Beirao da Veiga, F. Brezzi, L. D. Marini and A. Russo, Virtual element methods forgeneral second order elliptic problems on polygonal meshes, Math. Models Methods Appl. Sci. 26, 729–750 (2016).
[5] A. Cangiani, E. Georgoulis, T. Pryer, O. J. Sutton, A posteriori error estimates for the virtual element method, Numerische Mathematik, 137 (4), 857–893 (2017).
[6] L. Beirao da Veiga, C. Lovadina and G. Vacca, Divergence free virtual elements forthe Stokes problem on polygonal meshes, ESAIM : Math. Models Numer. Anal. 51, 509–535 (2017).
[7] F. Brezzi and L. D. Marini, Virtual element methods for plate bending problems, Comput. Methods Appl. Mech. Eng. 253, 455–462 (2013).

L. Beirao da Veiga, F. Brezzi, L. Marini, Virtual elements for linear elasticity problems, SIAM J. Numer. Anal. 51(2), 794–812 (2013)
L. Beirao da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtualelement method, arXiv:1607.05988.
M. F. Benedetto, S. Berrone, A. Borio, S. Pieraccini and S. Scialo, A hybrid mortarvirtual element method for discrete fracture network simulations, J. Comput. Phys. 306, 148–166 (2016).
M. F. Benedetto, S. Berrone, S. Pieraccini and S. Scialo, The virtual element methodfor discrete fracture network simulations, Comput. Methods Appl. Mech. Eng. 280,135–156 (2014).
F. Brezzi, R. S. Falk and L. D. Marini, Basic principles of mixed virtual elementmethods, ESAIM: Math. Model. Numer. Anal. 48, 1227–1240 (2014).
L. Beirao da Veiga, A. Chernov, L. Mascotto, A. Russo, Exponential convergence of the hpvirtual element method in presence of corner singularities. Numer. Math. 138(3), 581–613 (2018).
A.L. Gain, C. Talischi, G. H. Paulino, On the virtual element method for three-dimensionallinear elasticity problems on arbitrary polyhedral meshes, Comp. Methods Appl. Mech. Engrg. 282, 132–160 (2014).