184 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
One-parameter discontinuous Galerkin finite element discretisation of quasilinear parabolic problems
We consider the analysis of a one-parameter family of --version discontinuous Galerkin finite element methods for the numerical solution of quasilinear parabolic equations of the form u'-\na\cdot\set{a(x,t,\abs{\na u})\na u}=f(x,t,u) on a bounded open set \om\in\re^d, subject to mixed Dirichlet and Neumann boundary conditions on \pr\om. It is assumed that is a real--valued function which is Lipschitz-continuous and uniformly monotonic in its last argument, and is a real-valued function which is locally Lipschitz-continuous and satisfies a suitable growth condition in its last argument; both functions are measurable in the first and second arguments. For quasi--uniform --meshes, if u\in \H^1(0,T;\H^k(\om))\cap\L^\infty(0,T;\H^1(\om)) with , for discontinuous piecewise polynomials of degree not less than 1, the approximation error, measured in the broken norm, is proved to be the same as in the linear case: with
An hp-version discontinuous Galerkin method for integro-differential equations of parabolic type
We study the numerical solution of a class of parabolic integro-differential equations with weakly singular kernels. We use an -version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal -version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the -version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems
Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the 1-d wave equation
We perform a complete Fourier analysis of the semi-discrete 1-d wave equation
obtained through a P1 discontinuous Galerkin (DG) approximation of the
continuous wave equation on an uniform grid. The resulting system exhibits the
interaction of two types of components: a physical one and a spurious one,
related to the possible discontinuities that the numerical solution allows.
Each dispersion relation contains critical points where the corresponding group
velocity vanishes. Following previous constructions, we rigorously build wave
packets with arbitrarily small velocity of propagation concentrated either on
the physical or on the spurious component. We also develop filtering mechanisms
aimed at recovering the uniform velocity of propagation of the continuous
solutions. Finally, some applications to numerical approximation issues of
control problems are also presented.Comment: 6 pages, 2 figure
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