Characteristics of Mixed Finite Element Approximations


Characteristics of Mixed Finite Element Approximations
Tuesday, July 17, 2018
3:30PM – 5PM
POB 6.304

Phillippe R. B. Devloo

In this presentation we give an overview of the research efforts in developing mixed finite element approximations of conservation laws. We develop hp adaptive H(div) conforming spaces in one, two and three dimensions by combining vector fields with H^1 conforming spaces. By the fact that the vector fields are generated using Piola transformations the H(div) conforming spaces are applicable to two dimensional manifolds and/or nonlinear geometric maps. It is shown that by increasing the internal order of approximation of elements with order k boundary fluxes, convergence rates of order h^k+1 for flux and order H^k+2 for pressure are obtained.

Arbitrary orders of approximation H^k+n for div(σ) can be obtained by further increasing the internal order of approximation.
H(div) approximations can, similarly to H^1 approximations, benefit from the use of quarterpoint element mappings for the resolution of singularities.
H(div) approximations with internal bubble functions naturally lead to a procedure for computing highly efficient error estimators.

We combine H(div) approximations with a multiscale hybrid mixed (MHM) approximation method to obtain a multiscale approximation method with local conservation.

All numerical results were obtained by algorithms implemented in the NeoPZ programming environment that is freely available from github

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