An object-oriented framework for multiphysics problems combining different approximation spaces

An object-oriented framework is developed to implement discrete models in a monolithic solution of coupled systems of partial differential equations by combining different kinds of finite element approximation spaces chosen for each field. In this sense, we aim at contributing to applications of what is currently classified as “multiphysics simulation”, by composing numerical techniques where each physical phenomenon or scale component is approximated by its most appropriate numerical scheme. The integration of the methods is discussed, in a systematic and generic manner, based on an existing object oriented finite element computational framework, allowing any of the usual kinds of affine and/or curved element geometry (point, segment, triangle, quadrilateral, tetrahedral, hexahedral, prismatic or pyramidal). They can be used in several finite element formulations that require continuous, discontinuous, H(div)-conforming functions, or interactions with lower dimensional approximations, which typically occurs in hybrid methods or reduced models. Furthermore, to improve accuracy and/or efficiency, when necessary and/or allowed by the adopted mathematical formulation, different levels of mesh refinements can be adopted for each field, with different refinement configurations (h, p, hp, and directional refinements), as long as the meshes are nested. The generality and flexibility of the proposed framework is verified against a particular set of two dimensional test problems modeled by different formulations - Darcy's problem coupled with tracer transport, fluid flow coupled with geomechanical interaction, multiscale hybrid formulation for linear elasticity, and a hybrid mixed formulation for discrete fracture networks. The main implementation ideas can also be applied to three dimensional problems, as well as to other kinds of approximation space combinations.