Analysis and approximation of mixed-dimensional PDEs on 3D-1D domains coupled with Lagrange multipliers

by   Miroslav Kuchta, et al.

Coupled partial differential equations defined on domains with different dimensionality are usually called mixed dimensional PDEs. We address mixed dimensional PDEs on three-dimensional (3D) and one-dimensional domains, giving rise to a 3D-1D coupled problem. Such problem poses several challenges from the standpoint of existence of solutions and numerical approximation. For the coupling conditions across dimensions, we consider the combination of essential and natural conditions, basically the combination of Dirichlet and Neumann conditions. To ensure a meaningful formulation of such conditions, we use the Lagrange multiplier method, suitably adapted to the mixed dimensional case. The well posedness of the resulting saddle point problem is analyzed. Then, we address the numerical approximation of the problem in the framework of the finite element method. The discretization of the Lagrange multiplier space is the main challenge. Several options are proposed, analyzed and compared, with the purpose to determine a good balance between the mathematical properties of the discrete problem and flexibility of implementation of the numerical scheme. The results are supported by evidence based on numerical experiments.



There are no comments yet.


page 1

page 2

page 3

page 4


Abstractions and automated algorithms for mixed domain finite element methods

Mixed dimensional partial differential equations (PDEs) are equations co...

An arbitrary order Mixed Virtual Element formulation for coupled multi-dimensional flow problems

Discrete Fracture and Matrix (DFM) models describe fractured porous medi...

Numerical Approximations of Coupled Forward-Backward SPDEs

We propose and study a scheme combining the finite element method and ma...

Numerical valuation of Bermudan basket options via partial differential equations

We study the principal component analysis (PCA) based approach introduce...

Coupling non-conforming discretizations of PDEs by spectral approximation of the Lagrange multiplier space

This work focuses on the development of a non-conforming domain decompos...

Mixed-Dimensional Auxiliary Space Preconditioners

This work introduces nodal auxiliary space preconditioners for discretiz...

Numerical Analysis of a Parabolic Variational Inequality System Modeling Biofilm Growth at the Porescale

In this paper we consider a system of two coupled nonlinear diffusion–re...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.