Energy error estimates for the projection-difference method with the Crank--Nicolson scheme for parabolic equations (Q2760694)

Let \(V\subset H\) be a separable Hilbert space such that the embedding is compact and continuous. Let \(a(t,u,v)\) be a family of sesquilinear symmetric forms, where \(u\), \(v\in V\), \(t\in T\). It is assumed that the function \(t\to a(t,u,v)\) is absolutely continuous and the following conditions hold: NEWLINE\[NEWLINE \begin{gathered} | a(t,u,v)| \leq M_1\| u\|_V\| v\|_V, \\ a(t,u,u) \geq \delta\| u\| ^2_V,\quad \delta > 0, \\ \left| \frac{\partial}{\partial t}a(t,u,v)\right|\leq M_1\| u\|_V\| v\|_V. \end{gathered} NEWLINE\]NEWLINE The form \(a(t,u,v)\) generates a linear bounded operator \(A(t)\: v\to V'\) by the formula \(a(t,u,v) = (A(t)u,v)\). It is also assumed that there exist linear operators \(B(t)\: V\to H\) such that NEWLINE\[NEWLINE \| B(t)u\| M_3\| u\|_V\quad (u\in V) NEWLINE\]NEWLINE and the function \(t\to B(t)u\in H\) is measurable in \(t\in [0,T]\).NEWLINENEWLINE The aim of the article is to study, using approximate methods, the Cauchy problem for the abstract parabolic equation NEWLINE\[NEWLINE u'(t) + A(t)u(t) + B(t)u(t) = f(t),\quad u(0) = u^0\in V, NEWLINE\]NEWLINE where \(f\in L_2(0,T;H)\). The existence and uniqueness theorem for a weak solution was proven by \textit{V.~V.~Smagin} in [Differ. Equations. 32, No.~5, 723-725 (1996; Zbl 0884.34067)]. The author uses the projection-difference method with the Crank--Nicolson scheme in time to obtain the error estimates which guarantee the second-order convergence of the error to zero under minimal conditions on smoothness of the original problem.