Adaptive grid refinement is a critical component of the improvements that have recently been made in algorithms for the numerical solution of partial differential equations (PDEs). The development of new algorithms and computer codes for the solution of PDEs usually involves the use of proof-of-concept test problems. 2D elliptic problems are often used as the first test bed for new algorithms a...

We provide a detailed description of refinement in Event-B, both as a contribution in itself and as a foundation for the approach to simultaneous animation of multiple levels of refinement that we propose. We present an algorithm for simultaneous multi-level animation of refinement, and show how it can be used to detect a variety of errors that occur frequently when using refinement. The algori...

The fundamental solution of the Laplace equation in two dimensions is not contained in the Sobolev space H1(Ω) such that finite element error estimates are non-standard and quasi-uniform meshes are inappropriate. By using graded meshes L2-error estimates of almost optimal order are shown. As a by-product, we show for the Poisson equation with a right-hand side in L2 that appropriate mesh refine...

Abstract Non-uniform, dynamically adaptive meshes are a useful tool for reducing computational complexities geophysical simulations that exhibit strongly localised features such as is the case tsunami, hurricane or typhoon prediction. Using example of shallow water solver, this study explores set metrics to distinguish performance numerical methods using adaptively refined versus uniform indepe...

In the standard testing theory of DeNicola-Hennessy one process is considered to be a refinement of another if every test guaranteed by the former is also guaranteed by the latter. In the domain of web services this has been recast, with processes viewed as servers and tests as clients. In this way the standard refinement preorder between servers is determined by their ability to satisfy client...