On a full discretisation for nonlinear second-order evolution equations with monotone damping: construction, convergence, and error estimates
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 of the considered problem is proved, under some additional assumptions on the regularity of the a priori estimates and so on. The results obtained are applicable to the boundary value problem \[ \begin{cases} \partial_{tt}u + |\partial_t u|^{p-2} \partial_t u - \Delta u = f \;\text{in} \;\Omega \times (0,T), \\ u = 0 \;\text{on} \;\partial\Omega \times (0,T), \\ u(\cdot,0) = u_0, \quad \partial u_t u(\cdot,0) = v_0, \;\text{in} \;\Omega.\end{cases} \] At the end of the article, a numerical example for a model problem is considered.
scientific article; zbMATH DE number 6516359
scientific article