4,540 research outputs found
Stability of flat interfaces during semidiscrete solidification
No full textThe stability of flat interfaces with respect to a spatial semidiscretization of a solidification model is analyzed. The considered model is the quasi-static approximation of the Stefan problem with dynamical Gibbs--Thomson law. The stability analysis bases on an argument developed by Mullins and Sekerka for the undiscretized case. The obtained stability properties differ from those with respect to the quasi-static model for certain parameter values and relatively coarse meshes. Moreover, consequences on discretization issues are discussed
Efficient and Reliable A Posteriori Error Estimators for Elliptic Obstacle Problems
No full textA posteriori error estimators are derived for linear finite element approximations to elliptic obstacle problems. These estimators yield global upper and local lower bounds for the discretization error. Here discretization error means the sum of two contributions: the distance between continuous and discrete solution in the energy-norm and some quantity that is related to the distance of continuous and discrete contact set. Moreover, the local error indicators in the interior of the discrete contact set reduce to quantities that measure only data resolution
Adaptive Tree Approximation with Finite Element Functions: A First Look
No full textWe provide an introduction to adaptive tree approximation with finite element functions over meshes that are generated by bisection. This approximation technique can be seen as a benchmark for adaptive finite element methods, but may be also used therein for the approximation of data and coarsening. Correspondingly, we focus on approximation problems related to adaptive finite element methods, the design and performance of algorithms, and the resulting convergence rates, together with the involved regularity. For simplicity and clarity, these issues are presented and discussed in detail in the univariate case. The additional technicalities and difficulties of the multivariate case are briefly outlined
Quasi-optimal nonconforming methods for symmetric elliptic problems. II-Overconsistency and classical nonconforming elements
No full textWe devise variants of classical nonconforming methods for symmetric elliptic problems. These variants differ from the original ones only by transforming discrete test functions into conforming functions before applying the load functional. We derive and discuss conditions on these transformations implying that the ensuing method is quasi-optimal and that its quasi-optimality constant coincides with its stability constant. As applications, we consider the approximation of the Poisson problem with Crouzeix--Raviart elements and higher order counterparts and the approximation of the biharmonic problem with Morley elements. In each case, we construct a computationally feasible transformation and obtain a quasi-optimal method with respect to the piecewise energy norm on a shape regular mesh
Best Error Localizations for Piecewise Polynomial Approximation of Gradients, Functions and Functionals
No full textWe consider the approximation of (generalized) functions with continuous piecewise polynomials or with piecewise polynomials that are allowed to be discontinuous. Best error localization then means that the best error in the whole domain is equivalent to an appropriate accumulation of best errors in small domains, e.g., in mesh elements. We review and compare such best error localization in the three cases of the Sobolev-Hilbert triplet (H1 0 , L2 , H−1 )
The L2-projection and quasi-optimality of Galerkin methods for parabolic equations
No full textWe consider linear parabolic initial-boundary value problems and analyze Galerkin approximation in space. With the help of the inf-sup theory, we derive quasi-optimality results with respect to norms that arise from the standard weak formulation and from a formulation requiring only integrability in time. Moreover, we reveal that the H1-stability of the L2-projection is not only sufficient but also necessary for these results. As application, we consider conforming finite element approximation in space and derive a priori error bounds in terms of the local meshsize and piecewise regularity. The regularity is the minimal one indicated by approximation theory and matches regularity results for linear parabolic problems
Quasi-optimal nonconforming methods for symmetric elliptic problems. III-discontinuous Galerkin and other interior penalty methods
No full textWe devise new variants of the following nonconforming finite element methods: discontinuous Galerkin methods of fixed arbitrary order for the Poisson problem, the Crouzeix-Raviart interior penalty method for linear elasticity, and the quadratic C0 interior penalty method for the biharmonic problem. Each variant differs from the original method only in the discretization of the right-hand side. Before applying the load functional, a linear operator transforms nonconforming discrete test functions into conforming functions such that stability and consistency are improved. The new variants are thus quasi-optimal with respect to an extension of the energy norm. Furthermore, their quasi-optimality constants are uniformly bounded for shape regular meshes and tend to 1 as the penalty parameter increases
Quasi-optimal nonconforming methods for symmetric elliptic problems. I—abstract theory
No full textWe consider nonconforming methods for symmetric elliptic problems and characterize their quasi-optimality in terms of suitable notions of stability and consistency. The quasi-optimality constant is determined, and the possible impact of nonconformity on its size is quantified by means of two alternative consistency measures. Identifying the structure of quasi-optimal methods, we show that their construction reduces to the choice of suitable linear operators mapping discrete functions to conforming ones. Such smoothing operators are devised in the forthcoming parts of this work for various finite element spaces
Quasi-Optimal Nonconforming Methods for Second-Order Problems on Domains with Non-Lipschitz Boundary
No full textWe introduce new nonconforming finite element methods for elliptic problems of second order. In contrast to previous work, we consider mixed boundard conditions and the domain does not have to lie on one side of its boundary. Each method is quasi-optimal in a piecewise energy norm, thanks to the discretization of the load functional with a moment-preserving smoothing operator
Adaptive finite element methods
No full textThis is a survey of the theory of adaptive finite element methods (AFEMs), which are fundamental to modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and up-to-date discussion of AFEMs for linear second-order elliptic PDEs and dimension d > 1, with emphasis on foundational issues. After a brief review of functional analysis and basic finite element theory, including piecewise polynomial approximation in graded meshes, we present the core material for coercive problems. We start with a novel a posteriori error analysis applicable to rough data, which delivers estimators fully equivalent to the solution error. They are used in the design and study of three AFEMs depending on the structure of data. We prove linear convergence of these algorithms and rate-optimality provided the solution and data belong to suitable approximation classes. We also address the relation between approximation and regularity classes. We finally extend this theory to discontinuous Galerkin methods as prototypes of non-conforming AFEMs, and beyond coercive problems to inf-sup stable AFEMs
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