scholarly journals OptimalL ?-error estimates for piecewise linear finite elements in 1?n?3for 1?n?3

1987 ◽  
Vol 52 (5) ◽  
pp. 604-604
Author(s):  
Ludwig Kohaupt
Keyword(s):  
Finite Elements ◽  
Error Estimates ◽  
Piecewise Linear

scholarly journals Numerical analysis of sparse initial data identification for parabolic problems

2020 ◽  
Vol 54 (4) ◽  
pp. 1139-1180
Author(s):  
Dmitriy Leykekhman ◽  
Boris Vexler ◽  
Daniel Walter
Keyword(s):  
Numerical Analysis ◽  
Finite Elements ◽  
Initial Data ◽  
Error Estimates ◽  
Piecewise Linear ◽  
Control Variable ◽  
Parabolic Problems ◽  
Time Observation ◽  
Dirac Measures ◽  
Objective Functional

In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results.

scholarly journals International Workshop on Finite Elements for Microwave Engineering

2016 ◽  
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Dominant Role ◽  
Piecewise Linear ◽  
International Workshop ◽  
American Mathematical Society ◽  
Element Formulation ◽  
Microwave Engineering ◽  
History Of ◽  
Element Activity

When Courant prepared the text of his 1942 address to the American Mathematical Society for publication, he added a two-page Appendix to illustrate how the variational methods first described by Lord Rayleigh could be put to wider use in potential theory. Choosing piecewise-linear approximants on a set of triangles which he called elements, he dashed off a couple of two-dimensional examples and the finite element method was born. … Finite element activity in electrical engineering began in earnest about 1968-1969. A paper on waveguide analysis was published in Alta Frequenza in early 1969, giving the details of a finite element formulation of the classical hollow waveguide problem. It was followed by a rapid succession of papers on magnetic fields in saturable materials, dielectric loaded waveguides, and other well-known boundary value problems of electromagnetics. … In the decade of the eighties, finite element methods spread quickly. In several technical areas, they assumed a dominant role in field problems. P.P. Silvester, San Miniato (PI), Italy, 1992 Early in the nineties the International Workshop on Finite Elements for Microwave Engineering started. This volume contains the history of the Workshop and the Proceedings of the 13th edition, Florence (Italy), 2016 . The 14th Workshop will be in Cartagena (Colombia), 2018.

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

2020 ◽  
Vol 20 (2) ◽  
pp. 361-378
Author(s):  
Tamal Pramanick ◽  
Rajen Kumar Sinha
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Error Estimates ◽  
Finite Element Approximation ◽  
Parabolic Problems ◽  
Element Approximation ◽  
Dirichlet Problems ◽  
Element Mesh ◽  
Polygonal Domains ◽  
Numer Math

AbstractThe purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the {L^{\infty}(H^{1})}-norm and {L^{\infty}(L^{2})}-norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

scholarly journals On Superconvergence of a Gradient for Finite Element Methods for an Elliptic Equation with the Nonsmooth Right–hand Side

2002 ◽  
Vol 2 (3) ◽  
pp. 295-321 ◽  
Author(s):  
Alexander Zlotnik
Keyword(s):  
Finite Elements ◽  
Elliptic Equation ◽  
Error Estimates ◽  
Dirichlet Boundary ◽  
A Priori ◽  
Boundary Function ◽  
Right Hand ◽  
Priori Error Estimates ◽  
The Right ◽  
2D And 3D

AbstractThe elliptic equation under the nonhomogeneous Dirichlet boundary condition in 2D and 3D cases is solved. A rectangular nonuniform partition of a domain and polylinear finite elements are taken. For the interpolant of the exact solution u, a priori error estimates are proved provided that u possesses a weakened smoothness. Next error estimates are in terms of data. An estimate is established for the right–hand side f of the equation having a generalized smoothness. Error estimates are derived in the case of f which is not compatible with the boundary function. The proofs are based on some propositions from the theory of functions. The corresponding lower error estimates are also included; they justify the sharpness of the estimates without the logarithmic multipliers. Finally, we prove similar results in the case of 2D linear finite elements and a uniform partition.

Sign in / Sign up

Export Citation Format

Share Document