Adaptive techniques for time-dependent problems (Q686935)

The authors consider adaptive grid refinement strategies for time- dependent problems. As a sample problem, the 1-D heat conduction problem \(u_ t- u_{xx}= f\), \((x,t)\in (0,1)\times (0,T]\), with Dirichlet boundary conditions and known solutions, is studied in two cases: with ``global'' and ``local'' variation of the solutions \(u=\sin(2\pi x) \sin(2\pi t)\) and \(u=\exp(20t- 20t^ 2) \exp(-37x^ 2+ 66x- 30)\). Implicit Euler and Crank-Nicolson type methods with local refinements of the discretization grid -- in space as well as in time direction -- are proposed. At the interface between coarse and fine grid regions linear and quadratic interpolation is used in order to introduce auxiliary (slave) grid points. For \(C^ 4\)-solutions error bounds are given, generalizations to the 2-D case and variable coefficients are formulated. Numerical experiments show the effect of step size reduction. The influence of the type of the solution (local, global behavior), the relation between coarse space and time steps and of the interface interpolation order are studied in several error norms. The latter effect is proposed as a useful control for adaptive refinement. Conjectures, among others on unconditional stability and superconvergence effects, are formulated.