Modeling dimensionally-heterogeneous problems: analysis, approximation and applications

Abstract

In the present work a general theoretical framework for coupled dimensionally-heterogeneous partial differential equations is developed. This is done by recasting the variational formulation in terms of coupling interface variables. In such a general setting we analyze existence and uniqueness of solutions for both the continuous problem and its finite dimensional approximation. This approach also allows the development of different iterative substructuring solution methodologies involvingThe first author acknowledges the support of the Brazilian agencies CNPq and FAPERJ.

Augmented Interface Systems for the Darcy-Stokes Problem

“…This prevents us from guaranteeing their coerciveness (see [7]). In this section we present a different approach based on [5,3,4,6] consisting in writing the coupled Darcy-Stokes problem as a system of linear equations on Γ involving both variables λ and η.…”

Computational Reduction for Parametrized PDEs: Strategies and Applications

“…In particular, we consider the flow around a symmetric airfoil profile parametrized w.r.t. thickness µ 1 ∈ [4,24] and the angle of attack µ 2 ∈ [0.01, π/4]; the profile is rotated according to µ 2 , while the external boundaries remain fixed and the inflow velocity is parallel to the x axis (see Fig. 1).…”

“…Like in the usual DD framework, coupling conditions give rise to an interface problem (governed by the Steklov-Poincaré operator), which can be solved using classical tools derived from optimal control problems (giving rise to the so-called virtual control approach [15,18]). These techniques are suitable also for treating multiphysics problems, such as the coupling between hyperbolic and elliptic equations for boundary layers [18], or the Darcy-Stokes coupling for fluid flows through porous media [4].…”

Modeling Hemodynamics in Vascular Networks Using a Geometrical Multiscale Approach: Numerical Aspects

“…On the other hand, the mutual interactions between its compartments imply that these models should preferably not be considered separately. These requirements have led to the concept of geometrical multiscale modeling [7,13,18,20,24,48], where 6 the main idea is to couple dimensionally heterogeneous models, representing different physical compartments, to study the interaction between different geometrical scales. This approach applies 3D models only in those regions where a detailed knowledge of the flow field is needed, whereas 1D and 0D models are applied to represent the remaining part of the vascular tree.…”

“…To extent their use to non-periodic conditions, an approach for prescribing lumped parameter outflow boundary conditions that accommodate transient phenomena has been presented in [66]. Few groups have studied the coupling of 3D-1D-0D models to consider the interactions between the local and systemic circulation [7,10,20,40].…”