A mixed VEM scheme for a problem with edge and vertex singularities

ALEXIS JAWTUSCHENKO; ARIEL LOMBARDI

Santiago de Chile

Workshop; SANTIAGO NUMERICO III Ninth Workshop in Numerical Analysis and Partial Differential Equations; 2017

Pontificia Universidad Católica de Chile

We introduce and analyze a virtual element method for the mixed formulationof a Poisson problem with right--hand side inL2 and homogeneous Dirichlet conditionsin a non-convex polyhedral domain with edge and vertexsingularities, for which, in the presence of the mentionedsingularities, it is known that its solutionin general is not in H2. Asa consequence, the usual Finite Elements Methods are degraded andwe do not obtain an optimal convergence order in the general case.We present a VEM constructing a mesh that combines anisotropic prisms and tetrahedra with pyramids and avoids the use ofcertain tetrahedra that do not admitanisotropic estimates, recovering the optimal order of convergence.As stated in, if we make asubdivision of a general polyhedron $\Omega$ only withtetrahedra, then we do not obtain optimal error estimates with Mixed Raviart--Thomas Finite Elements for our problem. That is because there existsa class of anisotropic tetrahedra for whichanisotropic estimates needed in the analysis do not hold.For that reason we propose a method which among other things avoids the use of that kindof tetrahedra. In order to deal with general polyhedral domains we needto use mixed meshes, so we present a VEM scheme in a polyhedral meshmade oftetrahedra, triangular prisms and pyramids. This scheme can be seenas an extension of the method with classical lowest order Raviart--Thomaselements to the case in which the mesh contains pyramids. Besides,it is also an alternative to the generalization of the div--conformingelements on pyramids found for instance in, whose spaces, inparticular, contain rational functions.Incidentally, the number of mesh elements in our method is reducedby a constant factor.We show a discretization methodand introduce the corresponding discrete bilinear forms and show that the discrete problem is well posed byproving the discrete inf--sup condition.Next we prove that there exists a family of graded meshesT_h for which we have the optimal estimation |u-u_h| <= c h |f|, |p-p_h| <= c h |f|,with h <= (1/N)^{1/3}, where N is the number of elements of the mesh T_h.We show an example of a family of meshes for the Fichera domainthat verifies our hipothesis.