Abstract
We present a systematic description and comparison of the Finite Element Method (FEM) with the relatively new Virtual Element Method (VEM) for solving boundary value problems in linear elasticity, including primal and mixed formulations. The description highlights the common base and the essential difference between FEM and VEM: discretisation of the same primal (Galerkin) and mixed weak formulations and assembly of element-wise quantities, but different approaches to element shape functions. The mathematical formulations are complemented with detailed description of the computer implementation of all methods, including all versions of VEM, which will benefit readers willing to develop their own computational framework. Numerical solutions of several boundary value problems are also presented in order to discuss the weaker and stronger sides of the methods.
Highlights
Since its introduction in the 1950-s, the Finite Element Method (FEM) has become the most popular numerical method for solving boundary value problems in mechanics and physics
A parallel research direction, started in 2012 by a group of Italian mathematicians, is the Virtual Element Method (VEM) [41] with an implementation guide given in [42]. It is based on the same weak formulations as FEM, but inside each element the approximating local space is defined as the space of solutions of an elliptic partial differential equation (PDE)
In all methods we start with a quadrilateral mesh “Quad” which is triangulated to “Tri” for Primal finite element method (PFEM), MFEMWIS and the triangular version on Mixed virtual element method (MVEM) (“MVEM-Tri”)
Summary
Since its introduction in the 1950-s, the Finite Element Method (FEM) has become the most popular numerical method for solving boundary value problems in mechanics and physics. A detailed introduction can be found in the classical monograph on mixed finite elements [16], and for a general approach for developing stable discretisations, based on exterior calculus, we refer to [6] Both the primal and the mixed FEM are developed to deal with combinatorially regular (see Definition B.5) or semi-regular meshes, e.g., in 2D only triangular, only quadrilateral, and in some cases a combination of the two, elements are available. A parallel research direction, started in 2012 by a group of Italian mathematicians, is the Virtual Element Method (VEM) [41] with an implementation guide given in [42] It is based on the same weak formulations as FEM, but inside each element the approximating local space is defined as the space of solutions of an elliptic partial differential equation (PDE).
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