Results 11  20
of
57
A Robust Hierarchical Basis Preconditioner On General Meshes
 Numer. Math
, 1995
"... . In this paper, we introduce a multilevel direct sum space decomposition of general, possibly locally refined linear or multilinear finite element spaces. In contrast to the wellknown BPX and hierarchical basis preconditioners, the corresponding additive Schwarz preconditioner will be robust for ..."
Abstract

Cited by 17 (0 self)
 Add to MetaCart
. In this paper, we introduce a multilevel direct sum space decomposition of general, possibly locally refined linear or multilinear finite element spaces. In contrast to the wellknown BPX and hierarchical basis preconditioners, the corresponding additive Schwarz preconditioner will be robust for a class of singularly perturbed elliptic boundary value problems. Important for an efficient implementation is that stable bases of the subspaces defining our decomposition, consisting of functions having small supports can be easily constructed. 1. Background and motivation This paper deals with additive Schwarz multilevel preconditioners for solving symmetric second order linear elliptic boundary value problems (cf. [Xu92, Yse93, GO95a]). We assume a nested sequence of linear or multilinear finite element spaces M 0 ae M 1 ae : : : ae M J ae : : : and discretize the boundary value problem on M J using Galerkin's method. Additive Schwarz preconditioners are based on a subspace decomposi...
Hierarchical Error Estimator for Eddy Current Computation
 In ENUMATH99: Proceedings of the 3rd European Conference on Numerical Mathematics and Advanced Applictions
, 1999
"... We consider the quasimagnetostatic eddy current problem discretized by means of lowest order curlconforming finite elements (edge elements) on tetrahedral meshes. Bounds for the discretization error in the finite element solution are desirable to control adaptive mesh refinement. We propose a loca ..."
Abstract

Cited by 15 (2 self)
 Add to MetaCart
(Show Context)
We consider the quasimagnetostatic eddy current problem discretized by means of lowest order curlconforming finite elements (edge elements) on tetrahedral meshes. Bounds for the discretization error in the finite element solution are desirable to control adaptive mesh refinement. We propose a local aposteriori error estimator based on higher order edge elements: The residual equation is approximately solved in the space of phierarchical surpluses. Provided that a saturation assumption holds, we show that the estimator is both reliable and efficient.
A control reduced primal interior point method for a class of control constrained optimal control problems
 Computational Optimization and Applications
"... A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
A primal interior point method for control constrained optimal control problems with PDE constraints is considered. Pointwise elimination of the control leads to a homotopy in the remaining state and dual variables, which is addressed by a short step pathfollowing method. The algorithm is applied to the continuous, infinite dimensional problem, where discretization is performed only in the innermost loop when solving linear equations. The a priori elimination of the least regular control permits to obtain the required accuracy with comparatively coarse meshes. Convergence of the method and discretization errors are studied, and the method is illustrated at two numerical examples.
Domain decomposition operator splittings for the solution of parabolic equations
 SIAM J. Sci. Comput. (electronic
, 1998
"... Mathematics Faculty Publications by an authorized administrator of Wyoming Scholars Repository. For more information, please contact ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
Mathematics Faculty Publications by an authorized administrator of Wyoming Scholars Repository. For more information, please contact
A Twodimensional Moving Finite Element Method With Local Refinement Based On A Posteriori Error Estimates
 Applied Numer. Math
"... In this paper, we consider the numerical solution of time{dependent PDEs using a nite element method based upon rh{adaptivity. An adaptive horizontal method of lines strategy equipped with a posteriori error estimates to control the discretization through variable time steps and spatial grid adaptat ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
(Show Context)
In this paper, we consider the numerical solution of time{dependent PDEs using a nite element method based upon rh{adaptivity. An adaptive horizontal method of lines strategy equipped with a posteriori error estimates to control the discretization through variable time steps and spatial grid adaptations is used. Our approach combines an r{re nement method based upon solving so{called moving mesh PDEs with h{re nement. Numerical results are presented to demonstrate the capabilities and bene ts of combining mesh movement and local re nement.
AN ADAPTIVE WAVELET METHOD FOR THE CHEMICAL MASTER EQUATION
, 2010
"... An adaptive wavelet method for the chemical master equation is constructed. The method is based on the representation of the solution in a sparse Haar wavelet basis, the time integration by Rothe’s method, and an iterative procedure which in each timestep selects those degrees of freedom which are ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
(Show Context)
An adaptive wavelet method for the chemical master equation is constructed. The method is based on the representation of the solution in a sparse Haar wavelet basis, the time integration by Rothe’s method, and an iterative procedure which in each timestep selects those degrees of freedom which are essential for propagating the solution. The accuracy and efficiency of the approach is discussed, and the performance of the adaptive wavelet method is demonstrated by numerical examples.
KASKADE 3.0  An ObjectOriented Adaptive Finite Element Code
 International workshop
, 1995
"... KASKADE 3.0 was developed for the solution of partial differential equations in one, two, or three space dimensions. Its objectoriented implementation concept is based on the programming language C++ . Adaptive finite element techniques are employed to provide solution procedures of optimal comp ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
KASKADE 3.0 was developed for the solution of partial differential equations in one, two, or three space dimensions. Its objectoriented implementation concept is based on the programming language C++ . Adaptive finite element techniques are employed to provide solution procedures of optimal computational complexity. This implies a posteriori error estimation, local mesh refinement and multilevel preconditioning. The program was designed both as a platform for further developments of adaptive multilevel codes and as a tool to tackle practical problems. Up to now we have implemented scalar problem types like stationary or transient heat conduction. The latter one is solved with the Rothe method, enabling adaptivity both in space and time. Some nonlinear phenomena like obstacle problems or twophase Stefan problems are incorporated as well. Extensions to vectorvalued functions and complex arithmetic are provided. We have implemented several iterative solvers for both symmet...
A CACHEAWARE ALGORITHM FOR PDES ON HIERARCHICAL DATA STRUCTURES BASED ON SPACEFILLING CURVES
, 2006
"... Competitive numerical algorithms for solving partial differential equations have to work with the most efficient numerical methods like multigrid and adaptive grid refinement and thus with hierarchical data structures. Unfortunately, in most implementations, hierarchical data— typically stored in ..."
Abstract

Cited by 10 (1 self)
 Add to MetaCart
(Show Context)
Competitive numerical algorithms for solving partial differential equations have to work with the most efficient numerical methods like multigrid and adaptive grid refinement and thus with hierarchical data structures. Unfortunately, in most implementations, hierarchical data— typically stored in trees—cause a nonnegligible overhead in data access. To overcome this quandary— numerical efficiency versus efficient implementation—our algorithm uses spacefilling curves to build up data structures which are processed linearly. In fact, the only kind of data structure used in our implementation is stacks. Thus, data access becomes very fast—even faster than the common access to nonhierarchical data stored in matrices—and, in particular, cache misses are reduced considerably. Furthermore, the implementation of multigrid cycles and/or higher order discretizations as well as the parallelization of the whole algorithm become very easy and straightforward on these data structures.
Adaptive FEM for Reaction–Diffusion Equations
 INVITED TALK, GRID ADAPTATION IN COMPUTATIONAL PDES CONFERENCE, EDINBURGH 1996
, 1996
"... An integrated time–space adaptive finite element method for solving mixed systems of nonlinear parabolic, elliptic, and differential algebraic equations is presented. The approach is independent of the spatial dimension. For the discretization in time we use singly diagonally linearly implicit Ru ..."
Abstract

Cited by 9 (7 self)
 Add to MetaCart
An integrated time–space adaptive finite element method for solving mixed systems of nonlinear parabolic, elliptic, and differential algebraic equations is presented. The approach is independent of the spatial dimension. For the discretization in time we use singly diagonally linearly implicit Runge–Kutta methods of Rosenbrock type. Local time errors for the step size control are defined by an embedded strategy. A multilevel finite element Galerkin method is subsequently applied for the discretization in space. A posteriori estimates of local spatial discretization errors are obtained solving local problems with higher order approximation. Superconvergence arguments allow to simplify the required computations. Two different strategies to obtain the start grid of the multilevel process are compared. The devised method is applied to a solid–solid combustion problem.
Adaptive discrete Galerkin methods applied to the chemical master equation
, 2007
"... Abstract. In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signalling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equation, known as the c ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
(Show Context)
Abstract. In systems biology, the stochastic description of biochemical reaction kinetics is increasingly being employed to model gene regulatory networks and signalling pathways. Mathematically speaking, such models require the numerical solution of the underlying evolution equation, known as the chemical master equation (CME). Up to now, the CME has primarily been treated by MonteCarlo techniques, the most prominent of which is the stochastic simulation algorithm (Gillespie 1976). The paper presents an alternative, which focuses on the discrete partial differential equation (PDE) structure of the CME. This allows to adopt ideas from adaptive discrete Galerkin methods as first suggested in (Deuflhard, Wulkow 1989) for polyreaction kinetics and independently developed in (Engblom 2006). Among the two different options of discretizing the CME as a discrete PDE, Engblom had chosen the method of lines approach (first space, then time), whereas we strongly advocate to use the Rothe method (first time, then space) for clear theoretical and algorithmic reasons. Numerical findings at two rather challenging problems illustrate the promising features of the proposed method and, at the same time, indicate lines of necessary further improvement of the method worked out here.