In this work, we investigate (energy) stability of global radial basis function (RBF) methods for linear advection problems. Classically, boundary conditions (BC) are enforced strongly in RBF methods. By now it is well-known that this can lead to stability problems, however. Here, we follow a different path and propose two novel RBF approaches which are based on a weak enforcement of BCs. By using the concept of flux reconstruction and simultaneous approximation terms (SATs), respectively, we are able to prove that both new RBF schemes are strongly (energy) stable. Numerical results in one and two spatial dimensions for both scalar equations and systems are presented, supporting our theoretical analysis

In [2] is proposed a simplified DeC method, that, when combined with the residual distribution (RD) framework, allows to construct a high order, explicit FE scheme with continuous approximation avoiding the inversion of the mass matrix for hyperbolic problems. In this paper, we close some open gaps in the context of deferred correction (DeC) and their application within the RD framework. First, we demonstrate the connection between the DeC schemes and the RK methods. With this knowledge, DeC can be rewritten as a convex combination of explicit Euler steps, showing the connection to the strong stability preserving (SSP) framework. Then, we can apply the relaxation approach introduced in [22] and construct entropy conservative/dissipative DeC (RDeC) methods, using the entropy correction function proposed in [3].

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The Deferred Correction (DeC) methods combined with the residual distribution (RD) approach allow the construction of high order continuous Galerkin (cG) schemes avoiding the inversion of the mass matrix. With the application of entropy correction functions we can even obtain entropy conservative/dissipative spatial discretizations in this context. To handle entropy production in time, a relaxation approach has been suggested by Ketcheson. The main idea is to slightly modify the time-step size such that the approximated solution fulfills the underlying entropy conservation/dissipation constraint. In this paper, we first study the relaxation technique applied to the DeC approach as an ODE solver, then we extend this combination to the residual distribution method, requiring more technical steps. The outcome is a class of cG methods that is fully entropy conservative/dissipative and where we can still avoid the inversion of a mass matrix.

We present a class of discretisation spaces and H(div) - conformal elements that can be built on any polytope. Bridging the flexibility of the Virtual Element spaces towards the element's shape with the divergence properties of the Raviart - Thomas elements on the boundaries, the designed frameworks offer a wide range of H(div) - conformal discretisations. As those elements are set up through degrees of freedom, their definitions are easily amenable to the properties the approximated quantities are wished to fulfil. Furthermore, we show that one straightforward restriction of this general setting share its properties with the classical Raviart - Thomas elements at each interface, for any order and any polytopial shape. Then, to close the introduction of those new elements by an example, we investigate the shape of the basis functions corresponding to particular elements in the two dimensional case.