scholarly journals Anisotropic Error Estimates of the Linear Nonconforming Virtual Element Methods

2019 ◽  
Vol 57 (3) ◽  
pp. 1058-1081 ◽  
Author(s):  
Shuhao Cao ◽  
Long Chen
Keyword(s):  
Error Estimates ◽  
Virtual Element Methods ◽  
Virtual Element

Some Estimates for Virtual Element Methods

2017 ◽  
Vol 17 (4) ◽  
pp. 553-574 ◽  
Author(s):  
Susanne C. Brenner ◽  
Qingguang Guan ◽  
Li-Yeng Sung
Keyword(s):  
Error Estimates ◽  
Three Dimensions ◽  
Poisson Problem ◽  
Polyhedral Meshes ◽  
Virtual Element Methods ◽  
Virtual Element

AbstractWe present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions.

scholarly journals Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

2021 ◽  
Author(s):  
Andreas Dedner ◽  
Alice Hodson
Keyword(s):  
Fourth Order ◽  
Perturbation Parameter ◽  
Two Dimensions ◽  
Optimal Error Estimates ◽  
Projection Operators ◽  
Varying Coefficients ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Perturbation Problems ◽  
Fourth Order Problems

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

scholarly journals Bridging the hybrid high-order and virtual element methods

2020 ◽  
Author(s):  
Simon Lemaire
Keyword(s):  
High Order ◽  
Virtual Space ◽  
Approximation Properties ◽  
Virtual Spaces ◽  
Local Polynomial ◽  
Poisson Problem ◽  
Virtual Element Methods ◽  
Dimension 2 ◽  
Virtual Element ◽  
Polytopal Meshes

Abstract We present a unifying viewpoint on hybrid high-order and virtual element methods on general polytopal meshes in dimension $2$ or $3$, in terms of both formulation and analysis. We focus on a model Poisson problem. To build our bridge (i) we transcribe the (conforming) virtual element method into the hybrid high-order framework and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local virtual element discrete bilinear form is defined. This allows us to perform a unified analysis of virtual element/hybrid high-order methods, that differs from standard virtual element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

scholarly journals A posteriori error estimates for the virtual element method

2017 ◽  
Vol 137 (4) ◽  
pp. 857-893 ◽  
Author(s):  
Andrea Cangiani ◽  
Emmanuil H. Georgoulis ◽  
Tristan Pryer ◽  
Oliver J. Sutton
Keyword(s):  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
A Posteriori ◽  
Virtual Element Method ◽  
A Posteriori Error ◽  
Virtual Element ◽  
Element Method
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