We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier-St...

We discuss well-posedness and convergence theory for the DPG method applied to a general system of linear Partial Differential Equations (PDEs) and specialize the results to the classical Stokes problem. The Stokes problem is an iconic troublemaker for standard Bubnov Galerkin methods; if discretizations are not carefully designed, they may exhibit non-convergence or locking. By contrast, DPG d...

We present a new conservative multiscale method for Stokes flow in heterogeneous porous media. The method couples a discontinuous Galerkin finite element method (DG-FEM) at the macroscopic scale for the solution of an effective Darcy equation with a Stokes solver at the pore scale to recover effective permeabilities at macroscopic quadrature points. To avoid costly computation of numerous Stoke...

We propose and analyze a hybrid discontinuous Galerkin method for the solution 10 of incompressible flow problems, which allows to deal with pure Stokes, pure Darcy, and 11 coupled Darcy-Stokes flow in a unified manner. The flexibility of the method is demonstrated 12 in numerical examples.

In this paper, we extend the Bernardi-Raugel element [1] to convex polygonal meshes by using generalized barycentric coordinates. Comparing traditional discretizations defined on triangular and rectangular meshes, can be more flexible when dealing with complicated domains or curved boundaries. Theoretical analysis of new follows standard mixed finite theory for Stokes equations, i.e., shall pro...

In the articleMatrices de Stokes-Ramis et constantes de connexion pour les systèmes différentiels linéaires de niveau unique (P. Remy), we considered linear differential systems with a unique but arbitrary level and we stated formulae to express all the Stokes multipliers in terms of connection constants in the Borel plane generalizing thus the calculations made in the article Resurgence, Stoke...

We use the bivariate spline method to solve the time evolution Navier-Stokes equations numerically. The bivariate spline we use in this paper is the spline space of smoothness r and degree 3r over triangulated quadrangulations. The stream function formulation for the Navier-Stokes equations is employed. Galerkin's method is applied to discretize the space variables of the nonlinear fourth order...