Analysis and Approximation of Mixed-Dimensional PDEs on 3D-1D Domains Coupled with Lagrange Multipliers

Abstract

Coupled partial differential equations (PDEs) defined on domains with different dimensionality are usually called mixed-dimensional PDEs. We address mixed-dimensional PDEs on three-dimensional (3D) and one-dimensional (1D) domains, which gives rise to a 3D-1D coupled problem. Such a 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, which are 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 of determining 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.

Rational Approximations in Robust Preconditioning of Multiphysics Problems

“…The estimate (17) holds true if a method with optimal computational complexity is used to solve the auxiliary linear systems that appear in (15) or (16). In the multidimensional case, this means that such a preconditioned iterative solver is applied.…”

“…As it can be seen in [17], the standard finite element methods provide accurate error estimates in the norms of the introduced fractional order Sobolev spaces. The saddlepoint matrix corresponding to the obtained system of linear algebraic equations is sparse.…”

“…Let Ω ⊂ R 3 be a bounded domain, while Γ represents a 1D manifold (structure) inside Ω, see Figure 2. The following trace coupled problem is considered [17]: Here u, v, p are unknowns, the 1D manifold Γ is parameterized in terms of s, the Laplacian on Γ is understood in terms of the Laplace-Beltrami operator, T : Ω → Γ is a suitable trace operator, and pδ Γ is a Dirac measure such that Ω p(x)δ Γ w(x)dx = Γ p(s)δ Γ w(s)ds for a continuous function w.…”

Fluid-beam interaction: Capturing the effect of embedded slender bodies on global fluid flow and vice versa

“…For the case of mixed-dimensional problems, the form of such stable Lagrange multipliers is not yet well-studied. Alternatively, analogous to Nitsche's method for classical equal-dimensional embedded finite element problems [32,31], stabilized Lagrange multiplier methods can also be applied to mixed-dimensional embedded finite element problems [33,34].…”

“…In contrast to well-known trace theorems such as [35], which postulate existence of such a trace operator on smooth boundaries of codimension one, existence conditions on the restriction operator Π in the context of a greater dimensionality gap are not yet well-studied [36]. As one of the first publications addressing the lack of trace-type theorems for mixed-dimensional problems with codimension two, Kuchta et al [33] show sufficient regularity of such a restriction operator in the context of a mixed-dimensional model problem via averaging over a three-dimensional domain around the embedded manifold.…”