10 research outputs found
A posteriori error estimates in the maximum norm for parabolic problems
We derive a posteriori error estimates in the
norm for approximations of solutions to
linear para bolic equations. Using the elliptic reconstruction technique
introduced by Makridakis and Nochetto and heat kernel estimates for linear
parabolic pr oblems, we first prove a posteriori bounds in the maximum norm for
semidiscrete finite element approximations. We then establish a posteriori
bounds for a fully discrete backward Euler finite element approximation. The
elliptic reconstruction technique greatly simplifies our development by allow\
ing the straightforward combination of heat kernel estimates with existing
elliptic maximum norm error estimators
A Posteriori L_∞ (L_2 )+L_2 (H^1 )–Error Bounds in Discontinuous Galerkin Methods For Semidiscrete Semilinear Parabolic Interface Problems
ان الهدف من هذا البحث هو اشتقاق الخطأ البعدي لمسائل ذات الوسط البيني لشبه خطية متكافئة شبه متقطعة. وبشكل أكثر تحديدًا، تم تحليل الخطأ البعدي الأمثل لمسائل ذات الوسط البيني لشبه خطية متكافئة شبه متقطعة باستخدام تقنية إعادة الإهليلجية المقدمة من مارك داكس ونوكيتو (2003). الفكرة الأساسية لهذه التقنية هي استخدام مقدرات الأخطاء المشتقة من مشاكل الواجهة الإهليلجية للحصول على مقدرات مكافئ ذات ترتيب أمثل في المجال والزمان.The aim of this paper is to derive a posteriori error estimates for semilinear parabolic interface problems. More specifically, optimal order a posteriori error analysis in the - norm for semidiscrete semilinear parabolic interface problems is derived by using elliptic reconstruction technique introduced by Makridakis and Nochetto in (2003). A key idea for this technique is the use of error estimators derived for elliptic interface problems to obtain parabolic estimators that are of optimal order in space and time
A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\Omega})) in parabolic problems
We use the elliptic reconstruction technique in combination with a duality
approach to prove aposteriori error estimates for fully discrete back- ward
Euler scheme for linear parabolic equations. As an application, we com- bine
our result with the residual based estimators from the aposteriori esti- mation
for elliptic problems to derive space-error indicators and thus a fully
practical version of the estimators bounding the error in the L \infty (0, T ;
L2({\Omega})) norm. These estimators, which are of optimal order, extend those
introduced by Eriksson and Johnson (1991) by taking into account the error
induced by the mesh changes and allowing for a more flexible use of the
elliptic estima- tors. For comparison with previous results we derive also an
energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error
which simplifies a previous one given in Lakkis and Makridakis (2006). We then
compare both estimators (duality vs. energy) in practical situations and draw
conclusions.Comment: 30 pages, including 7 color plates in 4 figure
A Posteriori error control & adaptivity for Crank-Nicolson finite element approximations for the linear Schrodinger equation
We derive optimal order a posteriori error estimates for fully discrete
approximations of linear Schr\"odinger-type equations, in the
norm. For the discretization in time we use the Crank-Nicolson
method, while for the space discretization we use finite element spaces that
are allowed to change in time. The derivation of the estimators is based on a
novel elliptic reconstruction that leads to estimates which reflect the
physical properties of Schr\"odinger equations. The final estimates are
obtained using energy techniques and residual-type estimators. Various
numerical experiments for the one-dimensional linear Schr\"odinger equation in
the semiclassical regime, verify and complement our theoretical results. The
numerical implementations are performed with both uniform partitions and
adaptivity in time and space. For adaptivity, we further develop and analyze an
existing time-space adaptive algorithm to the cases of Schr\"odinger equations.
The adaptive algorithm reduces the computational cost substantially and
provides efficient error control for the solution and the observables of the
problem, especially for small values of the Planck constant
A comparison of duality and energy aposteriori estimates for L?(0,T;L2({\Omega})) in parabolic problems
We use the elliptic reconstruction technique in combination with a duality
approach to prove aposteriori error estimates for fully discrete back- ward
Euler scheme for linear parabolic equations. As an application, we com- bine
our result with the residual based estimators from the aposteriori esti- mation
for elliptic problems to derive space-error indicators and thus a fully
practical version of the estimators bounding the error in the L \infty (0, T ;
L2({\Omega})) norm. These estimators, which are of optimal order, extend those
introduced by Eriksson and Johnson (1991) by taking into account the error
induced by the mesh changes and allowing for a more flexible use of the
elliptic estima- tors. For comparison with previous results we derive also an
energy-based aposteriori estimate for the L \infty (0, T ; L2({\Omega}))-error
which simplifies a previous one given in Lakkis and Makridakis (2006). We then
compare both estimators (duality vs. energy) in practical situations and draw
conclusions.Comment: 30 pages, including 7 color plates in 4 figure
Gradient recovery in adaptive finite element methods for parabolic problems
We derive energy-norm aposteriori error bounds, using gradient recovery (ZZ)
estimators to control the spatial error, for fully discrete schemes for the
linear heat equation. This appears to be the first completely rigorous
derivation of ZZ estimators for fully discrete schemes for evolution problems,
without any restrictive assumption on the timestep size. An essential tool for
the analysis is the elliptic reconstruction technique.
Our theoretical results are backed with extensive numerical experimentation
aimed at (a) testing the practical sharpness and asymptotic behaviour of the
error estimator against the error, and (b) deriving an adaptive method based on
our estimators. An extra novelty provided is an implementation of a coarsening
error "preindicator", with a complete implementation guide in ALBERTA.Comment: 6 figures, 1 sketch, appendix with pseudocod
A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory
This paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial-boundary value problem of semilinear parabolic equations. The main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution. In the formulation used in this paper, the initial-boundary value problem is transformed into a fixed-point form using an analytic semigroup. The sufficient condition is derived from Banach\u27s fixed-point theorem. This paper also introduces a recursive scheme to extend a time interval in which the validity of the solution can be verified. As an application of this method, the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation
Recovery methods for evolution and nonlinear problems
Functions in finite dimensional spaces are, in general, not smooth enough to be differentiable in the classical sense and “recovered” versions of their first and second derivatives must be sought for certain applications. In this work we make use of recovered derivatives for applications in finite element schemes for two different purposes. We thus split this Thesis into two distinct parts.
In the first part we derive energy-norm aposteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error for fully discrete schemes of the linear heat equation. To our knowledge this is the first completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique introduced as an aposteriori analog to the elliptic (Ritz) projection.
Our theoretical results are backed up with extensive numerical experimentation aimed at (1) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (2) deriving an adaptive method based on our estimators.
An extra novelty is an implementation of a coarsening error “preindicator”, with a complete implementation guide in ALBERTA (versions 1.0–2.0).
In the second part of this Thesis we propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galërkin type using conforming finite elements and applied directly to the nonvariational(or nondivergence) form of a second order linear elliptic problem. The key tools are an
appropriate concept of the “finite element Hessian” based on a Hessian recovery and a Schur complement approach to solving the resulting linear algebra problem. The method
is illustrated with computational experiments on linear PDEs in nonvariational form.
We then use the nonvariational finite element method to build a numerical method for fully nonlinear elliptic equations. We linearise the problem via Newton’s method resulting in a sequence of nonvariational elliptic problems which are then approximated with the nonvariational finite element method. This method is applicable to general fully nonlinear PDEs who admit a unique solution without constraint.
We also study fully nonlinear PDEs when they are only uniformly elliptic on a certain class of functions. We construct a numerical method for the Monge–Ampère equation
based on using “finite element convexity” as a constraint for the aforementioned nonvariational finite element method. This method is backed up with numerical experimentation