Past

In the Said Business School

TBA

http://sag.maths.ox.ac.uk/samath/seminars/Sturm.pdf

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The use of discrete differential forms in the construction of finite element discretisations of the Sobolev spaces H^s, H(div) and H(curl) is now routinely applied by numerical analysts and engineers alike. However, little attention has been paid to the conditioning of the resulting stiffness matrices, particularly in the case of the non-uniform meshes that arise when adaptive refinement algorithms are used. We study this issue and show that the matrices are generally rather poorly conditioned. Typically, diagonal scaling is applied (often unwittingly) as a preconditioner. However, whereas diagonal scaling removes the effect of the mesh non-uniformity in the case of Sobolev spaces H^s, we show this is not so in the case of the spaces H(curl) and H(div). We trace the reason behind this difference, and give a simple remedy for curing the problem.

TBA

We propose a new parallel domain decomposition algorithm to solve symmetric linear systems of equations derived from the discretization of PDEs on general unstructured grids of triangles or tetrahedra. The algorithm is based on a single-level Schwarz alternating procedure and a modified conjugate gradient solver. A single layer of overlap has been adopted in order to simplify the data-structure and minimize the overhead. This approach makes the global convergence rate vary slightly with the number of domains and the algorithm becomes highly scalable. The algorithm has been implemented in Fortran 90 using MPI and hence portable to different architectures. Numerical experiments have been carried out on a SunFire 15K parallel computer and have been shown superlinear performance in some cases.

In S18