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57
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 70 (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
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The cascadic multigrid method for elliptic problems
, 1996
"... The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without ..."
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Cited by 54 (5 self)
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The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, which used the CG method as basic iteration method on each level. In [18] it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasiuniform triangulations. The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly nonuniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method.
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 51 (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 cruci ..."
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Cited by 50 (8 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.
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 ..."
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Cited by 49 (7 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.
An Outline of Adaptive Wavelet Galerkin Methods for Tikhonov Regularization of Inverse Parabolic Problems
, 2002
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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 turn ..."
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Cited by 22 (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
Linearly implicit time discretization of nonlinear parabolic equations.
 IMA J. Numer. Anal.,
, 1995
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