291 research outputs found
Discontinuous Galerkin Methods for the Biharmonic Problem on Polygonal and Polyhedral Meshes
We introduce an -version symmetric interior penalty discontinuous
Galerkin finite element method (DGFEM) for the numerical approximation of the
biharmonic equation on general computational meshes consisting of
polygonal/polyhedral (polytopic) elements. In particular, the stability and
-version a-priori error bound are derived based on the specific choice of
the interior penalty parameters which allows for edges/faces degeneration.
Furthermore, by deriving a new inverse inequality for a special class {of}
polynomial functions (harmonic polynomials), the proposed DGFEM is proven to be
stable to incorporate very general polygonal/polyhedral elements with an
\emph{arbitrary} number of faces for polynomial basis with degree . The
key feature of the proposed method is that it employs elemental polynomial
bases of total degree , defined in the physical coordinate
system, without requiring the mapping from a given reference or canonical
frame. A series of numerical experiments are presented to demonstrate the
performance of the proposed DGFEM on general polygonal/polyhedral meshes
Discontinuous Galerkin finite element approximation of quasilinear elliptic boundary value problems II: Strongly monotone quasi-Newtonian flows
In this article we develop both the a priori and a posteriori error analysis of hp–version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain = 2,3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp–adaptive refinement algorithm
Solving elliptic eigenvalue problems on polygonal meshes using discontinuous Galerkin composite finite element methods
In this paper we introduce a discontinuous Galerkin method on polygonal meshes. This method arises from the discontinuous Galerkin composite finite element method (DGFEM) for source problems on domains with micro-structures. In the context of the present paper, the flexibility of DGFEM is applied to handle polygonal meshes. We prove the a priori convergence of the method for both eigenvalues and eigenfunctions for elliptic eigenvalue problems. Numerical experiments highlighting the performance of the proposed method for problems with discontinuous coefficients and on convex and non-convex polygonal meshes are presented
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