A least-squares virtual element method for second-order elliptic problems

2020 ◽  
Vol 80 (8) ◽  
pp. 1873-1886
Author(s):  
Ying Wang ◽  
Gang Wang
Keyword(s):  
Least Squares ◽  
Elliptic Problems ◽  
Second Order ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Second Order Elliptic Problems ◽  
Element Method

Virtual Element Method for general second-order elliptic problems on polygonal meshes

2016 ◽  
Vol 26 (04) ◽  
pp. 729-750 ◽  
Author(s):  
L. Beirão da Veiga ◽  
F. Brezzi ◽  
L. D. Marini ◽  
A. Russo
Keyword(s):  
Variable Coefficients ◽  
Second Order ◽  
Projection Operators ◽  
Virtual Element Method ◽  
Basic Principles ◽  
Order Elliptic Operator ◽  
Smooth Coefficients ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Element Method

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214]), we use here, in a systematic way, the [Formula: see text]-projection operators as designed in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391]. In particular, the present method does not reduce to the original Virtual Element Method of [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the case of variable coefficients produces, in general, sub-optimal results.

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