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36
A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations
 475 (1994) MR 94j:65136
"... Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for spacetime finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizat ..."
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Cited by 48 (2 self)
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Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for spacetime finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the socalled θscheme, which includes the implicit and explicit Euler methods and the CrankNicholson scheme. 1.
Adaptive Multilevel Methods in Three Space Dimensions
 Int. J. Numer. Methods Eng
, 1993
"... this paper to collect wellknown results on 3D mesh refinement ..."
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Cited by 42 (6 self)
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this paper to collect wellknown results on 3D mesh refinement
Parallel Multigrid in an Adaptive PDE Solver Based on Hashing and SpaceFilling Curves
, 1997
"... this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 ..."
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Cited by 39 (3 self)
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this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 deals with the partitioning and distribution of adaptive grids with spacefilling curves and section 5 gives the main features of our new parallelized adaptive multilevel solver. In section 6 we present the results of numerical experiments on a parallel cluster computer with up to 64 processors. It is shown that our approach works nicely also for problems with severe singularities which result in locally refined meshes. Here, the work overhead for load balancing and data distribution remains only a small fraction of the overall work load. 2. DATA STRUCTURES FOR ADAPTIVE PDE SOLVERS 2.1. Adaptive Cycle
A Posteriori Error Estimates for Elliptic Problems in Two and Three Space Dimensions
 SIAM J. Numer. Anal
, 1993
"... . Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a cruc ..."
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Cited by 39 (5 self)
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. Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A wellknown class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations. Key words: adaptive finite element methods, aposteriori error estimates AMS (MOS) subject classifications: 65N30, 65N50, 65N55, 35J25 1 submi...
Residual Based A Posteriori Error Estimators For Eddy Current Computation
, 1999
"... We consider H(curl;\Omega\Gamma3932/608 problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori err ..."
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Cited by 27 (6 self)
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We consider H(curl;\Omega\Gamma3932/608 problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori 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 fundamental tool in the numerical analysis is a Helmholtztype decomposition of the error into an irrotational part and a weakly solenoidal part.
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 ..."
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Cited by 16 (0 self)
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. 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...
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 ..."
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Cited by 10 (7 self)
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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.
Discrete Transparent Boundary Conditions for the Numerical Solution of Fresnel's Equation
 Computers Math. Applic
, 1993
"... The paper presents a construction scheme of deriving transparent , i. e. reflectionfree, boundary conditions for the numerical solution of Fresnel's equation (being formally equivalent to Schrodinger's equation). These boundary conditions appear to be of a nonlocal Cauchy type. As it turns out, eac ..."
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Cited by 9 (3 self)
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The paper presents a construction scheme of deriving transparent , i. e. reflectionfree, boundary conditions for the numerical solution of Fresnel's equation (being formally equivalent to Schrodinger's equation). These boundary conditions appear to be of a nonlocal Cauchy type. As it turns out, each kind of linear implicit discretization induces its own discrete transparent boundary conditions. Key words. Fresnel equation, boundary condition, adaptive Rothe method Contents
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 ..."
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Cited by 9 (1 self)
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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.
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 ..."
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Cited by 8 (0 self)
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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...