2,951 research outputs found
A posteriori error control for discontinuous Galerkin methods for parabolic problems
We derive energy-norm a posteriori error bounds for an Euler time-stepping
method combined with various spatial discontinuous Galerkin schemes for linear
parabolic problems. For accessibility, we address first the spatially
semidiscrete case, and then move to the fully discrete scheme by introducing
the implicit Euler time-stepping. All results are presented in an abstract
setting and then illustrated with particular applications. This enables the
error bounds to hold for a variety of discontinuous Galerkin methods, provided
that energy-norm a posteriori error bounds for the corresponding elliptic
problem are available. To illustrate the method, we apply it to the interior
penalty discontinuous Galerkin method, which requires the derivation of novel a
posteriori error bounds. For the analysis of the time-dependent problems we use
the elliptic reconstruction technique and we deal with the nonconforming part
of the error by deriving appropriate computable a posteriori bounds for it.Comment: 6 figure
Space Decompositions and Solvers for Discontinuous Galerkin Methods
We present a brief overview of the different domain and space decomposition
techniques that enter in developing and analyzing solvers for discontinuous
Galerkin methods. Emphasis is given to the novel and distinct features that
arise when considering DG discretizations over conforming methods. Connections
and differences with the conforming approaches are emphasized.Comment: 2 pages 2 figures no table
An agglomeration-based massively parallel non-overlapping additive Schwarz preconditioner for high-order discontinuous Galerkin methods on polytopic grids
In this article we design and analyze a class of two-level non-overlapping
additive Schwarz preconditioners for the solution of the linear system of
equations stemming from discontinuous Galerkin discretizations of second-order
elliptic partial differential equations on polytopic meshes. The preconditioner
is based on a coarse space and a non-overlapping partition of the computational
domain where local solvers are applied in parallel. In particular, the coarse
space can potentially be chosen to be non-embedded with respect to the finer
space; indeed it can be obtained from the fine grid by employing agglomeration
and edge coarsening techniques. We investigate the dependence of the condition
number of the preconditioned system with respect to the diffusion coefficient
and the discretization parameters, i.e., the mesh size and the polynomial
degree of the fine and coarse spaces. Numerical examples are presented which
confirm the theoretical bounds
Stabilised finite element methods for ill-posed problems with conditional stability
In this paper we discuss the adjoint stabilised finite element method
introduced in, E. Burman, Stabilized finite element methods for nonsymmetric,
noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on
Scientific Computing, and how it may be used for the computation of solutions
to problems for which the standard stability theory given by the Lax-Milgram
Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to
ill-posed problems that have some conditional stability property and prove
(conditional) error estimates in an abstract framework. As a model problem we
consider the elliptic Cauchy problem and provide a complete numerical analysis
for this case. Some numerical examples are given to illustrate the theory.Comment: Accepted in the proceedings from the EPSRC Durham Symposium Building
Bridges: Connections and Challenges in Modern Approaches to Numerical Partial
Differential Equation
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