A Gagliardo – Nirenberg inequality , with application to duality - based a posteriori error estimation in the L 1 norm
نویسندگان
چکیده
A Gagliardo–Nirenberg inequality, with application to duality-based a posteriori error estimation in the L 1 norm Dedicated to Professor Boško S. Jovanovi´c on the occasion of his sixtieth birthday Endre Süli We establish the Gagliardo–Nirenberg-type multiplicative interpolation inequality v L 1 (R n) ≤ Cv 1/2 Lip (R n) v 1/2 BV(R n) ∀v ∈ BV(R n), where C is a positive constant, independent of v. We then use a local version of this inequality to derive an a posteriori error bound in the L 1 (Ω) norm, with ¯ Ω ⊂ Ω = (0, 1) n , for a finite-element approximation to a boundary value problem for a first-order linear hyperbolic equation, under the limited regularity requirement that the solution to the problem belongs to BV(Ω).
منابع مشابه
A Gagliardo–nirenberg Inequality, with Application to Duality-based a Posteriori Estimation in the L Norm
We prove the Gagliardo–Nirenberg-type multiplicative interpolation inequality ‖v‖L1(Rn) ≤ C‖v‖ 1/2 Lip′(Rn)‖v‖ 1/2 BV(Rn) ∀v ∈ Lip ′(Rn) ∩ BV(R), where C is a positive constant, independent of v. Here ‖·‖Lip′(Rn) is the norm of the dual to the Lipschitz space Lip 0(R) := C 0,1 0 (Rn) = C0,1(Rn) ∩C0(R) and ‖ · ‖BV(Rn) signifies the norm in the space BV(Rn) consisting of functions of bounded vari...
متن کاملGagliardo–nirenberg Inequalities with a Bmo Term
We give a simple direct proof of the interpolation inequality ‖∇f‖2 L2p C‖f‖BMO‖f‖W 2, p , where 1 < p < ∞. For p = 2 this inequality was obtained by Meyer and Rivière via a different method, and it was applied to prove a regularity theorem for a class of Yang–Mills fields. We also extend the result to higher derivatives, sharpening all those cases of classical Gagliardo– Nirenberg inequalities...
متن کاملOptimal L-Riemannian Gagliardo-Nirenberg inequalities
Let (M, g) be a compact Riemannian manifold of dimension n ≥ 2 and 1 < p ≤ 2. In this work we prove the validity of the optimal Gagliardo-Nirenberg inequality M |u| r dv g p rθ ≤ A opt M |∇u| p g dv g + B M |u| p dv g M |u| q dv g p(1−θ) θq for a family of parameters r, q and θ. Our proof relies strongly on a new distance lemma which holds for 1 < p ≤ 2. In particular, we obtain Riemannian vers...
متن کاملEstimation of Tensor Truncations Estimation of Tensor Truncations
Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to l or L norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require l ∞ or L∞ error estimates. Due to the huge dimensions of the tensor spaces, a direct estim...
متن کاملA Posteriori Error Estimates for Parabolic Problems via Elliptic Reconstruction and Duality
We use the elliptic reconstruction technique in combination with a duality approach to prove a posteriori error estimates for fully discrete backward Euler scheme for linear parabolic equations. As an application, we combine our result with the residual based estimators from the a posteriori estimation for elliptic problems to derive space-error estimators and thus a fully practical version of ...
متن کامل