Hermite methods, as introduced by Goodrich et al. in [15], combine Hermite interpolation and staggered (dual) grids to produce stable high order accurate schemes for the solution of hyperbolic PDEs. We introduce three variations of this Hermite method which do not involve time evolution on dual grids. Computational evidence is presented regarding stability, high order convergence, and dispersion/dissipation properties for each new method. Hermite methods may also be coupled to discontinuous Galerkin (DG) methods for additional geometric flexibility [4]. An example illustrates the simplification of this coupling for Hermite methods.

We describe and review non oscillatory residual distribution schemes that are rather natural extension of high order finite volume schemes when a special emphasis is put on the structure of the computational stencil. We provide their connections with standard stabilized finite element and discontinuous Galerkin schemes, show that their are really non oscillatory. We also discuss the extension to these methods to parabolic problems. We also draw some research perspectives.

We propose and analyze a domain decomposition method on non-matching gridsfor partial differential equations with non-negativecharacteristic form. No weak or strong continuity of the finiteelement functions, their normal derivatives, or linearcombinations of the two is imposed across the boundaries of the subdomains.Instead, we employ suitable bilinear forms defined on the commoninterfaces, typical of discontinuous Galerkinapproximations.We prove an error bound which is optimal with respect to the mesh–size andsuboptimal with respect to the polynomial degree.Our analysis is valid for arbitrary shape–regular meshes and arbitrarypartitions into subdomains.Our method can be applied to advective, diffusive, and mixed–type equations,as well,and is well-suited for problems coupling hyperbolic and elliptic equations.We present some two-dimensionalnumerical results that support our analysis for the case of linear finite elements.