We study the two-phase Stokes flow driven by surface tension with two fluids of equal viscosity, separated by an asymptotically flat interface with graph geometry. The flow is assumed to be two-dimensional with the fluids filling the entire space. We prove well-posedness and parabolic smoothing in Sobolev spaces up to critical regularity. The main technical tools are an analysis of nonlinear singular integral operators arising from the hydrodynamic single-layer potential and abstract results on nonlinear parabolic evolution equations.

In this paper, we present and study a mixed variational method in order to approximate,with the finite element method, a Stokes problem with Tresca friction boundary conditions.These non-linear boundary conditions arise in the modeling of mold filling process bypolymer melt, which can slip on a solid wall. The mixed formulation is based on adualization of the non-differentiable term which define the slip conditions. Existence anduniqueness of both continuous and discrete solutions of these problems is guaranteed bymeans of continuous and discrete inf-sup conditions that are proved. Velocity and pressureare approximated by P1 bubble-P1 finite element and piecewise linearelements are used to discretize the Lagrange multiplier associated to the shear stress onthe friction boundary. Optimal a priori error estimates are derived usingclassical tools of finite element analysis and two uncoupled discrete inf-sup conditionsfor the pressure and the Lagrange multiplier associated to the fluid shear stress.

We introduce a piecewise P2-nonconforming quadrilateralfinite element. First, we decompose a convex quadrilateral into the union of fourtriangles divided by its diagonals. Then the finite element space is defined by the set ofall piecewise P2-polynomials that are quadratic in eachtriangle and continuously differentiable on the quadrilateral. The degrees of freedom(DOFs) are defined by the eight values at the two Gauss points on each of the four edgesplus the value at the intersection of the diagonals. Due to the existence of one linearrelation among the above DOFs, it turns out the DOFs are eight. Global basis functions aredefined in three types: vertex-wise, edge-wise, and element-wise types. The correspondingdimensions are counted for both Dirichlet and Neumann types of elliptic problems. Forsecond-order elliptic problems and the Stokes problem, the local and global interpolationoperators are defined. Also error estimates of optimal order are given in both brokenenergy and L2(Ω) norms. The proposed elementis also suitable to solve Stokes equations. The element is applied to approximate eachcomponent of velocity fields while the discontinuousP1-nonconforming quadrilateral element is adopted toapproximate the pressure. An optimal error estimate in energy norm is derived. Numericalresults are shown to confirm the optimality of the presented piecewiseP2-nonconforming element on quadrilaterals.

We construct a finite volume element method based on the constrained nonconforming rotated Q1-constant element (CNRQ1-P0) for the Stokes problem. Two meshes are needed, which are the primal mesh and the dual mesh. We approximate the velocity by CNRQ1 elements and the pressure by piecewise constants. The errors for the velocity in the H1 norm and for the pressure in the L2 norm are O(h) and the error for the velocity in the L2 norm is O(h2). Numerical experiments are presented to support our theoretical results.

The penalty method when applied to the Stokes problem provides a very efficient algorithm for solving any discretization of this problem since it gives rise to a system of two equations where the unknowns are uncoupled. For a spectral or spectral element discretization of the Stokes problem, we prove a posteriori estimates that allow us to optimize the penalty parameter as a function of the discretization parameter. Numerical experiments confirm the interest of this technique.

We present and analyse in this paper a novel cell-centered collocated finite volume scheme for incompressible flows.Its definition involves a partition of the set of control volumes; each element of this partition is called a cluster and consists in a few neighbouring control volumes.Under a simple geometrical assumption for the clusters, we obtain that the pair of discrete spaces associating the classical cell-centered approximation for the velocities and cluster-wide constant pressures is inf-sup stable; in addition, we prove that a stabilization involving pressure jumps only across the internal edges of the clusters yields a stable scheme with the usual collocated discretization (i.e. , in particular, with control-volume-wide constant pressures), for the Stokes and the Navier-Stokes problem.An analysis of this stabilized scheme yields the existence of the discrete solution (and uniqueness for the Stokes problem). The convergence of the approximate solution toward the solution to the continuous problem as the mesh size tends to zero is proven, provided, in particular, that the approximation of the mass balance flux is second order accurate; this condition imposes some geometrical conditions on the mesh.Under the same assumption, an error analysis is provided for the Stokes problem: it yields first-order estimates in energy norms.Numerical experiments confirm the theory and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

We consider a family of quadrilateral or hexahedral mixed hp-finite elements for an incompressible flow problem with Q r -elements for the velocity anddiscontinuous $P_{r-1}$ -elements for the pressure where the orderr can vary from element to elementbetween 2 and an arbitrary bound.For multilevel adaptive grids with hanging nodes and a sufficiently small mesh size,we prove the inf-sup condition uniformly with respect to the meshsize and the polynomial degree.

We present and analyse in this paper a novel colocated Finite Volume scheme for the solution of the Stokes problem.It has been developed following two main ideas.On one hand, the discretization of the pressure gradient term is built as the discrete transposed of the velocity divergence term, the latter being evaluated using a natural finite volume approximation; this leads to a non-standard interpolation formula for the expression of the pressure on the edges of the control volumes.On the other hand, the scheme is stabilized using a finite volume analogue to the Brezzi-Pitkäranta technique.We prove that, under usual regularity assumptions for the solution (each component of the velocity in H2(∞) and pressure in H1(∞)), the scheme is first order convergent in the usual finite volume discrete H1 norm and the L2 norm for respectively the velocity and the pressure, provided, in particular, that the approximation of the mass balance flux is of second order.With the above-mentioned interpolation formulae, this latter condition is satisfied only for particular meshes: acute angles triangulations or rectangular structured discretizations in two dimensions, and rectangular parallelepipedic structured discretizations in three dimensions.Numerical experiments confirm this analysis and show, in addition, a second order convergence for the velocity in a discrete L2 norm.

We analyze a new formulation of the Stokes equations in three-dimensional axisymmetric geometries, relying on Fourier expansion with respect to the angular variable: the problem for each Fourier coefficient is two-dimensional and has six scalar unknowns, corresponding to the vector potential and the vorticity. A spectral discretization is built on this formulation, which leads to an exactly divergence-free discrete velocity. We prove optimal error estimates.

For the Stokes problem in a two- or three-dimensionalbounded domain, we propose a new mixed finite element discretization which relies ona nonconforming approximation of the velocity and a more accurate approximation of thepressure. We prove that the velocity and pressure discrete spaces are compatible, in thesense that they satisfy an inf-sup condition of Babuška and Brezzi type, and wederive some error estimates.

The LBB condition is well-known to guarantee the stability of a finiteelement (FE) velocity - pressure pair in incompressible flow calculations.To ensure the condition to be satisfied a certain constant should be positive andmesh-independent. The paper studies the dependence of the LBB condition on thedomain geometry. For model domains such as strips and rings thesubstantial dependence of this constant on geometry aspect ratios is observed.In domains with highly anisotropic substructures this may require special care with numerics to avoid failures similar to those whenthe LBB condition is violated. In the core of the paperwe prove that for any FE velocity-pressure pair satisfying usual approximationhypotheses the mesh-independent limit in the LBB condition is not greater thanits continuous counterpart, the constant from the Nečas inequality.For the latter the explicit and asymptotically accurate estimates are proved. The analytic results are illustrated by several numerical experiments.