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This paper provides an overview of the new features of the finite element library deal.II, version 9.3.

Convergence of a finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. Well-posedness is obtained in [J. Numer. Math., 29(2), 2021]. Theoretical convergence is proved via a compactness argument. The numerical phase saturation converges strongly to a weak solution in L 2 in space and in time whereas the numerical phase pressures converge strongly to weak solutions in L 2 in space almost everywhere in time. The proof is not straightforward because of the degeneracy of the phase mobilities and the unboundedness of the derivative of the capillary pressure.

We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz–Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between the role of the discretized operators and that of the domain meshing. Various algebraic strategies are discussed for the solution of the resulting matrix equation.

The paper studies a higher order unfitted finite element method for the Stokes system posed on a surface in ℝ 3 . The method employs parametric P k - P k −1 finite element pairs on tetrahedral bulk mesh to discretize the Stokes system on embedded surface. Stability and optimal order convergence results are proved. The proofs include a complete quantification of geometric errors stemming from approximate parametric representation of the surface. Numerical experiments include formal convergence studies and an example of the Kelvin–Helmholtz instability problem on the unit sphere.