88 research outputs found
On the Convergence of Adaptive Iterative Linearized Galerkin Methods
A wide variety of different (fixed-point) iterative methods for the solution
of nonlinear equations exists. In this work we will revisit a unified iteration
scheme in Hilbert spaces from our previous work that covers some prominent
procedures (including the Zarantonello, Ka\v{c}anov and Newton iteration
methods). In combination with appropriate discretization methods so-called
(adaptive) iterative linearized Galerkin (ILG) schemes are obtained. The main
purpose of this paper is the derivation of an abstract convergence theory for
the unified ILG approach (based on general adaptive Galerkin discretization
methods) proposed in our previous work. The theoretical results will be tested
and compared for the aforementioned three iterative linearization schemes in
the context of adaptive finite element discretizations of strongly monotone
stationary conservation laws
Fully Adaptive Newton-Galerkin Methods for Semilinear Elliptic Partial Differential Equations
In this paper we develop an adaptive procedure for the numerical solution of
general, semilinear elliptic problems with possible singular perturbations. Our
approach combines both a prediction-type adaptive Newton method and an adaptive
finite element discretization (based on a robust a posteriori error analysis),
thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical
experiments underline the robustness and reliability of the proposed approach
for different examples
A Nitsche Finite Element Approach for Elliptic Problems with Discontinuous Dirichlet Boundary Conditions
We present a numerical approximation method for linear diffusion-reaction
problems with possibly discontinuous Dirichlet boundary conditions. The
solution of such problems can be represented as a linear combination of
explicitly known singular functions as well as of an -regular part. The
latter part is expressed in terms of an elliptic problem with regularized
Dirichlet boundary conditions, and can be approximated by means of a Nitsche
finite element approach. The discrete solution of the original problem is then
defined by adding the singular part of the exact solution to the Nitsche
approximation. In this way, the discrete solution can be shown to converge of
second order with respect to the mesh size
Lockingâfree DGFEM for elasticity problems in polygons
The hâversion of the discontinuous Galerkin finite element method (hâDGFEM) for nearly incompressible linear elasticity problems in polygons is analysed. It is proved that the scheme is robust (lockingâfree) with respect to volume locking, even in the absence of H2âregularity of the solution. Furthermore, it is shown that an appropriate choice of the finite element meshes leads to robust and optimal algebraic convergence rates of the DGFEM even if the exact solutions do not belong to H
Computing the Entropy of a Large Matrix
Given a large real symmetric, positive semidefinite m-by-m matrix, the goal
of this paper is to show how a numerical approximation of the entropy, given by
the sum of the entropies of the individual eigenvalues, can be computed in an
efficient way. An application from quantum-optics illustrates the new
algorithm
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