On a full discretisation for nonlinear second-order evolution equations with monotone damping: construction, convergence, and error estimates
type: Item
instance of: scholarly article
title: On a full discretisation for nonlinear second-order evolution equations with monotone damping: construction, convergence, and error estimates
zbMATH Open document ID: 1334.65094
author: Etienne Emmrich
author: David Šiška
author: Mechthild Thalhammer
published in: Foundations of Computational Mathematics
publication date: 2015-12-04T00:00:00Z
full work available at URL: On a Full Discretisation for Nonlinear Second Order Evolution Equations with Monotone Damping Construction Convergence and Error Estimates.pdf
review text: The article deals with the following initial value problem for a nonlinear evolution equation of second order in time: \[ \begin{cases} u'' + Au' + Bu = f, \quad t \in (0,T), \\ u(0) = u_0, \;\;u'(0) = v_0.\end{cases} \] Here, it is assumed that the nonlinear operator \(:\;V_A \to V_A^*\) is hemicontinuous, monotone, coercive with exponent \(p \geq 2\) and satisfies a suitable growth condition with exponent \(p - 1\), the operator \(B:\;V_B \to V_B^*\) is linear, bounded, symmetric, and strongly positive; \(V_A\) and \(V_B\) are real, reflexive, and separable Banach spaces, which are dense as well as continuously embedded in a Hilbert space \(H\), so that \[ V_A \cap V_B = V \subset V_A,\, V_B \subset H = H^* \subset V_A^*, V_B^* \subset V^* = V_!^* + V_B^*. \] Under these conditions, for the initial problem under consideration, there exists a unique weak solution \(u \in L^\infty(0,T;V_B)\) such that \(u' \in L^\infty(0,T;H) \cap L^p(0,T;V_A)\) and \(u'' \in L^{p^*}(0,T;V_A^*) + L^\infty(0,T;V_B^*) \subset L^{p^*}(0,T;V^*)\). For finding the discrete solution \((v^n)_{n=1}^N\) to this initial problem, the following difference scheme (full discretisation) is considered: \[ (D_2v^n|\varphi) + \langle Av^n|\varphi \rangle_{V_A^* \times V_A} + \langle B(u^0 + K_\tau v^n)|\varphi \rangle_{V_B^* \times V_B} = \langle f^n|\varphi \rangle_{V_A^* \times V_A}, \] where \[ K_\tau v^n = \tau \sum_{j=1}^n v^j + \widetilde{K}_\tau v^n, \quad \widetilde{K}_\tau v^n = \frac12 \, (1 - 3^{-n+1})\tau v^1 - \tau \sum_{j=2}^n 3^{-n-1+j} v^j, \] \[ (D_1v^1|\varphi)_H + \langle Av^1|\varphi \rangle_{V_A^* \times V_A} + \langle B(v^0 + K_\tau v^1)|\varphi \rangle_{V_B^* \times V_B} = \langle f^1|\varphi\rangle_{V_A^* \times V_A}, \] \[ \bigg(D_1u^1 = u^1 - u^0, \quad D_2u^n = \tau^{-1}\bigg(\frac32 u^n - 2u^{n-1} + \frac12 u^{n-2}\bigg), \] \[ 0 = t_0 < \ldots < t_n = n\tau < \ldots < t_N = T, \quad \tau = \frac{T}N\bigg). \] The convergence of the discrete solutions to the weak solution …
Mathematics Subject Classification ID: 65J08
Mathematics Subject Classification ID: 65L05
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 47J35
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 35K90
zbMATH DE Number: 6516359
zbMATH Keywords: nonlinear evolution equation
zbMATH Keywords: monotone operators
zbMATH Keywords: weak solutions
zbMATH Keywords: time and spacial discretization
zbMATH Keywords: convergence
zbMATH Keywords: error estimates
zbMATH Keywords: initial value problem
zbMATH Keywords: Banach space
zbMATH Keywords: Hilbert space
zbMATH Keywords: difference scheme
zbMATH Keywords: numerical example
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
OpenAlex ID: W2078598047
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DOI: 10.1007/S10208-014-9238-4
description: scientific article; zbMATH DE number 6516359