Optimal spatial error estimates for DG time discretizations (Q2863115)

The article deals with the following linear Cauchy problem NEWLINE\[NEWLINEu'(t) + Au(t) = g(t) \;\;(0 < t < T), \;\;\;u(0) = u^0,NEWLINE\]NEWLINE where \(u \in C^1(0,T,X)\), \(u', g \in C(0,T,H)\), \(u^0 \in X\), \(X \subset H\) are Hilbert spaces (\(X\) is dense in \(H\)), \(A:\;X \to X^*\) is a linear self-adjoint positive definite operator. The approximate solution \(u_h(t) \in C^1(0,T,X_h)\) (\(X_h\) is a Hilbert space, \(X_h \subset X\)) to this problem is defined as functions \(U \in X_h^\tau\) satisfying the discrete linear problem NEWLINE\[NEWLINE\int_{I_m} ((U',v) + (AU,v)) \, dt + (\{U\}_{m-1},v_+^{m-1}) = \int_{I_m} (g,v) \, dt, \;\;\forall v \in X_h^\tau, \;\forall m = 1,\dots,r,NEWLINE\]NEWLINE NEWLINE\[NEWLINE(U_-^0,v) (u^0,v) \;\;\forall v \in X_h;NEWLINE\]NEWLINE here are \(I_m = (t_{m-1},t_m)\), \(0 = t_0 < t_1 < \dots < t_r = T\), \(\tau = \max \{t_m - t_{m-1}\}\), \(X_h^\tau\) is the space of piecewise polynomial functions: NEWLINE\[NEWLINEX_h^\tau = \bigg\{v \in L^2(0,T,X_h):\;v\big|_{I_m} = \sum_{j=0}^q v_{j,m}t^j, \;v_{j,m} \in X_h\bigg\},NEWLINE\]NEWLINE \(v_\pm^m = \lim\limits_{t \to t_m \pm} v(t)\), \(\{v\}_m = v_+^m - v_-^m\). The main result is the following estimate NEWLINE\[NEWLINE\max_{m=1,\dots,r} \;\sup_{I_m} \;\|U - u\| \leq C\bigg(\sup_{(0,T)} \|R_hu - u\| + \sup_{(0,T)} \|R_hu' - u'\| + \tau^{q+1} + \|U_-^0 - u^0\|\bigg)NEWLINE\]NEWLINE (\(R_h:\;X \to X_h\) is the Riesz projection), where \(C\) depends on \(u\) and \(T\), but is independent of \(h\) and \(\tau\). The abstract theorem is applied to the heat and the semi-linear convection-diffusion equations.