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

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