A priori error analysis for HDG methods using extensions from subdomains to achieve boundary conformity

A Hybridizable Discontinuous Galerkin solver for the Grad–Shafranov equation

“…One of the advantages of the definition (8e) is that it keeps equations (8a) and (8b) completely in terms of local quantities (once the hybrid unknown has been determined). It has been shown in the linear case (in [35] for polyhedral domains and in [34] for curved domains) that the method achieves optimal convergence order when τ is kept of order one. In our computations the value of the stabilization parameter was set to τ = 1.…”

“…However, the main drawback of standard unfitted methods is that only low order approximations can be obtained due to the fact that the boundary data on the computational domain is imposed "away" from the true boundary. Recently, an approach that combines the flexibility in the generation of the mesh characteristic of unfitted methods with a technique to transfer the boundary data from Γ to Γ h has been developed [28,34]. This method proposes an approximation of the boundary data by performing line integration along segments, called transferring paths, connecting Γ h to Γ.…”

Conforming and nonconforming virtual element methods for elliptic problems

“…An alternative approach is to completely relax the inter-element conformity requirements for the discrete space, so that simple (polynomial) spaces can be used. For instance, discontinuous Galerkin and weak Galerkin methods, whereby inter-element continuity is weakly imposed, are naturally suited to general meshes; see [15,17,28] and the references therein.…”

eXtended Hybridizable Discontinous Galerkin (X-HDG) for Void Problems

“…Therefore, the mesh should be adapted to properly describe the interface geometry, requiring continuous remeshing in the case of evolving interfaces. On other hand, a methodology for the solution of elliptic prob-lems with meshes not fitting the boundary is proposed in [8,9]. The solution at the boundary is extrapolated from nodal values of the computational mesh; consequently, some restrictive requirements on the distance from the computational mesh to the boundary are necessary to achieve optimal convergence, limiting the practical applicability of the proposal.…”

“…Differently to [8,9], here the computational mesh always covers the domain and, therefore, no extrapolations are required, leading to a more robust method. In fact, in 2D, the X-HDG method proposed here is formally equivalent to a standard HDG method applied on a cut mesh combining triangles and quadrilateral elements with a P k polynomial approximation (i.e.…”