We present, on the simplest possible case, what we consider as the very basic features of the (brand new) Virtual Element Methods. As the readers will easily see, the V.E.M. could easily be regarded as the ultimate evolution of the Mimetic Finite Difference approach. However, in this last step they became so close to the traditional Finite Element Methods that we decided to use a different perspective and a different name. Now the V.E. spaces are just like the usual Finite Element spaces with the addition of suitable non polynomial functions. This is far from being a new idea. The trick here is to take the spaces and the degrees of freedom in such a way that the elementary stiffness matrix can be computed without actually computing these non polynomial functions, but just using the degrees of freedom. In doing that we can easily deal with complicated element geometries and/or higher order continuity conditions (like C^1, C^2, etc.). The idea is quite general, and could be applied to a number of different situations and problems, and several papers are being written, dealing with different cases. Here however we want to be as clear as possible, and to present the simplest possible case that still gives the flavor of the whole idea.
Basic principles of Virtual Element Methods
MARINI, LUISA DONATELLA;
2013-01-01
Abstract
We present, on the simplest possible case, what we consider as the very basic features of the (brand new) Virtual Element Methods. As the readers will easily see, the V.E.M. could easily be regarded as the ultimate evolution of the Mimetic Finite Difference approach. However, in this last step they became so close to the traditional Finite Element Methods that we decided to use a different perspective and a different name. Now the V.E. spaces are just like the usual Finite Element spaces with the addition of suitable non polynomial functions. This is far from being a new idea. The trick here is to take the spaces and the degrees of freedom in such a way that the elementary stiffness matrix can be computed without actually computing these non polynomial functions, but just using the degrees of freedom. In doing that we can easily deal with complicated element geometries and/or higher order continuity conditions (like C^1, C^2, etc.). The idea is quite general, and could be applied to a number of different situations and problems, and several papers are being written, dealing with different cases. Here however we want to be as clear as possible, and to present the simplest possible case that still gives the flavor of the whole idea.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
