Dual-consistent approximations of PDEs and super-convergent functional output

Finite-difference operators satisfying the summation-by-parts (SBP) rule can be used to obtain high-order accurate time-stable schemes for partial differential equations (PDEs), when the boundary and interface conditions are imposed weakly by the simultaneous approximation term method (SAT). The most common SBP-SAT discretizations are accurate of order 2p in the interior and order p close to the boundaries, which yields a global accuracy of p+ 1 for first order PDEs. However, Berg and Nordström [2012] and Hicken and Zingg [2014] have shown that any linear functional computed from the time-dependent numerical solution will be accurate of order 2p if the spatial discretization is dual-consistent. This text discusses the concepts of dual problem and dual consistency. We walk through some of the definitions and results in Berg and Nordström [2012] and Hicken and Zingg [2014] while emphasizing the underlying functional analysis considerations. 1 Dual problems and dual consistency We introduce the concepts of dual problem and dual consistency by considering linear partial differential equations with homogeneous boundary conditions. Let L be a linear differential operator of order m on a domain Ω and consider the problem Lu− f = 0, x ∈ Ω, (1.1) ∗Division of Scientific Computing, Department of Information Technology, Uppsala university, SE-751 05 Uppsala, Sweden. [email protected]