3,969 research outputs found
hp-Version discontinuous Galerkin finite element methods for semilinear parabolic problems
We consider the hp-version interior penalty discontinuous Galerkin finite element method (hp-DGFEM) for semilinear parabolic equations with mixed Dirichlet and Neumann boundary conditions. Our main concern is the error analysis of the hp--DGFEM on shape--regular spatial meshes. We derive error bounds under various hypotheses on the regularity of the solution, for both the symmetric and non--symmetric versions of DGFEM
Discontinuous Galerkin Methods for Mass Transfer through Semi-Permeable Membranes
A discontinuous Galerkin (dG) method for the numerical solution of
initial/boundary value multi-compartment partial differential equation (PDE)
models, interconnected with interface conditions, is presented and analysed.
The study of interface problems is motivated by models of mass transfer of
solutes through semi-permeable membranes. More specifically, a model problem
consisting of a system of semilinear parabolic advection-diffusion-reaction
partial differential equations in each compartment, equipped with respective
initial and boundary conditions, is considered. Nonlinear interface conditions
modelling selective permeability, congestion and partial reflection are applied
to the compartment interfaces. An interior penalty dG method is presented for
this problem and it is analysed in the space-discrete setting. The a priori
analysis shows that the method yields optimal a priori bounds, provided the
exact solution is sufficiently smooth. Numerical experiments indicate agreement
with the theoretical bounds and highlight the stability of the numerical method
in the advection-dominated regime
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
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