Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm

We consider the approximation in the reaction-diffusion norm with continuous finite elements and prove that the best error is equivalent to a sum of the local best errors on pairs of elements. The equivalence constants do not depend on the ratio of diffusion to reaction. As application, we derive local error functionals that ensure robust performance of adaptive tree approximation in the reaction-diffusion norm.Comment: 21 pages, 1 figur

A Posteriori Error Estimates for Nonlinear Problems. L r (0,T; L rho (Omega))-Error Estimates for Finite Element Discretizations of Parabolic Equations

Error Estimates Let X,Y be two Banach spaces with norms #.# X and #.# Y . For any element u # X and any real number R > 0 set BX (u, R) := {v # X : #u - v#X < R}. We denote by L(X, Y ) and Isom(X, Y ) # L(X, Y ) the Banach space of continuous linear maps of X in Y equipped with the operator norm #.# L(X,Y ) and the open subset of linear homeomorphisms of X onto Y . By Y # := L(Y, IR) and < ., . > Y we denote the dual space of Y and the corresponding duality pairing. Finally, A # # L(Y # , X # ) denotes the adjoint of a given operator A # L(X, Y ). Let F # C 1 (X, Y # ) be a given continuously di#erentiable function. Given a solution u 0 # X of problem (1.1) and an arbitrary element u # X "close" to u 0 , we may estimate the error #u - u 0 #X by the residual #F (u)# Y # (cf. Proposition 2.1 in [10]). For parabolic pde's we thus obtain control on the L r (0, T ; W 1,# 0 (#))-norm of the error (cf. Section 3 for the definition of these spaces and ..

Non-Overlapping Domain Decomposition Methods Interpreted as Multiplicative Subspace Correction Algorithms

Setting We retain the notation of the previous section. Now, we will consider particular splittings V = V 1 +V 2 . To this end let W be another finite dimensional Hilbert-space with scalar product < ., . >. We assume that V and W are coupled by a continuous and surjective trace operator # : V # W . In the introductory example, e.g., V and W correspond to finite dimensional approximations of H 1 0 (#) and H 1/2 00 (#). # is the standard trace operator which maps a function defined on # onto its restriction to #. Set U 1 := ker(#) = {v # V : #(v) = 0} and U 2 := U #A 1 = {v # V : (Av, w) = 0 #w # U 1 }. Denote by E : W # V the maximal right inverse of # which is defined by E# = argmin v## -1 (#) #v# 2 1 ## # W. Another characterization of E#, # # W , is given by #(E#) = # and (A(E#), v) = 0 #v # U 1 . (4.1) In the introductory example, e.g., E corresponds to the harmonic extension of functions on #. The characterization (4.1) implies U 2 = E(W ). Thus, we h..

Primer of adaptive finite element methods

Adaptive finite element methods (AFEM) are a fundamental numerical instrument in science and engineering to approximate partial differential equations. In the 1980s and 1990s a great deal of effort was devoted to the design of a posteriori error estimators, following the pioneering work of Babuska. These are computable quantities, depending on the discrete solution(s) and data, that can be used to assess the approximation quality and improve it adaptively. Despite their practical success, adaptive processes have been shown to converge, and to exhibit optimal cardinality, only recently for dimension d > 1 and for linear elliptic PDE. These series of lectures presents an up-to-date discussion of AFEM encompassing the derivation of upper and lower a posteriori error bounds for residual-type estimators, including a critical look at the role of oscillation, the design of AFEM and its basic properties, as well as a complete discussion of convergence, contraction property and quasi-optimal cardinality of AFEM