Mixed virtual element methods for general second order elliptic problems on polygonal meshes

Abstract

In the present paper we introduce a Virtual Element Method (VEM) for the approximate solution of general linear second order elliptic problems in mixed form, allowing for variable coefficients. We derive a theoretical convergence analysis of the method and develop a set of numerical tests on a benchmark problem with known solution.

Serendipity face and edge VEM spaces

“…In other words, ϕ will be a scalar bubble of degree k + 1. Recalling the results of [7] we can define now η E as the minimum number of different planes necessary to cover ∂E, and deduce that if η E > k + 1 then Z k is reduced to {0}, and the degrees of freedom (8.31)-(8.33) and (8.25) will already be able to identify all the elements of S k in a unique way; this would mean that we can take S = M in (3.1). Otherwise, for η E ≤ k + 1, we will have that the dimension of Z k is equal to π k+1−η,3 , and we need an S such that S − M ≥ π k+1−η,3 .…”

“…So far so good. Now, to the polygon E we attach the integer number η E defined as (5.4) η E := the minimum number of straight lines necessary to cover ∂E, and we recall the following obvious but useful property (already used in [7]). …”

“…Always following what has been done in [7], we can distinguish different types of strategies to be adopted in coding these elements, in particular when dealing with very general geometries. The two extremes of this set of possible choices have been called the stingy choice and the lazy choice.…”

“…This justifies the effort to eliminate some internal degrees of freedom and, for H(curl)-conforming polyhedrons, also some of the degrees of freedom internal to faces. We are doing this here, following a Serendipity-like strategy, in the stream of what has been done, for instance, in [3], [7].…”

BDDC and FETI-DP for the virtual element method

“…An implicit knowledge of the local shape functions allows the evaluation of the operators and matrices needed in the method. Its implementation is described in [7] and the p and hp versions of the method are discussed and analyzed in [4,12,51]. Despite its recent introduction, VEM has already been applied and extended to study a wide variety of different model problems.…”