Exact estimates for the error gradient of locally one-dimensional methods for multidimensional equation of heat conduction (Q1390382)

It is known since the sixties that locally one-dimensional difference methods have rather weak accuracy even for problems with smooth solutions in the case of the Dirichlet boundary conditions dependent of \(f\). The authors consider homogeneous boundary conditions but the right-hand term is \(f\in L_2(Q_T)\). They show that a condition of type \(\tau= O(h^2)\) is necessary for convergence in the natural energy norm. Estimates of type \(O(\tau^{1/2}+h)\) are derived under special conditions on \(f\). They provide also an interesting example, connected with absence of convergence in the energy norm under the condition \(\tau\geq \varepsilon h\).