10 research outputs found
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
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
A note on the design of hp-version interior penalty discontinuous Galerkin finite element methods for degenerate problems
We consider a variant of the hp-version interior penalty discontinuous Galerkin finite element method (IP-DGFEM) for second order problems of degenerate type. We do not assume uniform ellipticity of the diffusion tensor. Moreover, diffusion tensors or arbitrary form are covered in the theory presented. A new, refined recipe for the choice of the discontinuity-penalisation parameter (that is present in the formlation of the IP-DGFEM) is given. Making use of the recently introduced augmented Sobolev space framework, we prove an hp-optimal error bound in the energy norm and an h-optimal and slightly p-suboptimal (by only half an order of p) bound in the L2 norm, provided that the solution belongs to an augmented Sobolev space
Poincaré-type inequalities for broken Sobolev spaces
We present two versions of general Poincaré-type inequalities for functions in broken Sobolev spaces, providing bounds for the Lq-norm of a function in terms of its broken H1-norm
hp-version discontinuous Galerkin finite element methods for nonlinear parabolic problems
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