scholarly journals Adaptive Non-Hierarchical Galerkin Methods for Parabolic Problems with Application to Moving Mesh and Virtual Element Methods

2021 ◽  
Author(s):  
Andrea Cangiani ◽  
Emmanuil H. Georgoulis ◽  
Oliver J. Sutton
Keyword(s):  
Moving Mesh ◽  
Parabolic Problems ◽  
Galerkin Methods ◽  
Virtual Element Methods ◽  
Virtual Element

Recent techniques for PDE discretizations on polyhedral meshes

2014 ◽  
Vol 24 (08) ◽  
pp. 1453-1455 ◽  
Author(s):  
N. Bellomo ◽  
F. Brezzi ◽  
G. Manzini
Keyword(s):  
Discontinuous Galerkin ◽  
Finite Differences ◽  
Discontinuous Galerkin Methods ◽  
Galerkin Methods ◽  
Special Issue ◽  
Polyhedral Meshes ◽  
Recent Developments ◽  
Virtual Element Methods ◽  
Mimetic Finite Differences ◽  
Virtual Element

This brief paper is an introduction to the papers published in a special issue devoted to survey on recent techniques for discretizing Partial Differential Equations on general polygonal and polyhedral meshes. The number of different techniques to deal with discretizations on polygonal and polyhedral meshes is quite huge, and their history is quite long. Here we concentrate on the most recent techniques, including Mimetic Finite Differences, Virtual Element Methods, and the recent developments, in this direction, of Finite Volumes and Discontinuous Galerkin Methods.

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.

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.

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