We discuss a family of discontinuous Petrov–Galerkin (DPG) schemes for quite general partial differential operators. The starting point of our analysis is the DPG method introduced by [Demkowicz , 49 (2011) 1788–1809; Zitelli , 230 (2011) 2406–2432]. This discretization results in a sparse positive definite linear algebraic system which can be obtained from a saddle point problem by an element-wise Schur complement reduction applied to the test space. Here, we show that the abstract framework of...
We consider mixed finite element discretizations of second order elliptic boundary value problems. Emphasis is on the efficient iterative solution by multilevel techniques with respect to an adaptively generated hierarchy of nonuniform triangulations. In particular, we present two multilevel solvers, the first one relying on ideas from domain decomposition and the second one resulting from mixed hybridization. Local refinement of the underlying triangulations is done by efficient and reliable a...
A new Schwarz method for nonlinear systems is presented, constituting
the multiplicative variant of a straightforward additive scheme.
Local convergence can be guaranteed under suitable assumptions.
The scheme is applied to nonlinear acoustic-structure interaction problems.
Numerical examples validate the theoretical results. Further improvements are
discussed by means of introducing overlapping subdomains and employing an inexact
strategy for the local solvers.
Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We focus on mortar finite element methods on non-matching triangulations. In particular, we discuss and analyze dual Lagrange multiplier spaces for lowest order finite elements. These non standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a consequence, standard efficient iterative...
Domain decomposition techniques provide a powerful tool for the numerical
approximation of partial differential equations.
We focus on mortar finite element methods on non-matching triangulations.
In particular, we discuss and analyze dual Lagrange multiplier spaces
for lowest order finite elements.
These non standard Lagrange multiplier spaces yield optimal discretization
schemes and a locally supported basis for the associated
constrained mortar spaces. As a consequence,
standard efficient iterative...
We consider ;Ω)-elliptic problems that have been discretized by
means of Nédélec's edge elements on tetrahedral meshes. Such
problems
occur in the numerical computation of eddy currents. From the defect
equation we derive localized expressions that can be used
as error estimators to control adaptive
refinement.
Under certain assumptions on material parameters and computational
domains, we derive local lower bounds and a global upper bound for the
total error measured in the energy norm. The...
Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces in case of hexahedral triangulations. As a result, standard efficient iterative solvers as...
Domain decomposition techniques provide a flexible tool for the numerical
approximation of partial differential equations. Here, we consider
mortar techniques for quadratic finite elements in 3D with
different Lagrange multiplier spaces.
In particular, we
focus on Lagrange multiplier spaces
which yield optimal discretization
schemes and a locally supported basis for the associated
constrained mortar spaces in case
of hexahedral triangulations. As a result,
standard efficient iterative solvers...
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