# Lecture SS 20 Selected Topics in Scientific Computing

## Practical Numerical Methods for PDEs

Requirements: Wissenschaftliches Rechnen/Scientific Computing I or II (V3E1/2/F4E1), basic programming skills in C

In this two-hour lecture without exercises, we will study computational techniques to solve partial differential equations on adaptive meshes. Adaptivity is often necessary to resolve areas of interest in a larger domain to increased accuracy, or to resolve moving phenomena in a time-stepping simulation. Using highly resolved meshes in turn motivates to consider parallel implementations, that is, numerical solutions using a network of computers with distributed local memory. In this case, selected data about both the mesh and the simulation must be made accessible by one computer for access by another.

An efficient and scalable computational approach to support such methods is that of an adaptive octree, or multiple such octrees connected to an adaptive forest. Here, each element of the mesh corresponds bijectively to a leaf of the forest. Establishing a total order on the leaves establishes a space-filling curve that has attractive properties with respect to the computational performance. We will review the fundamental concepts, definitions and algorithms and study how we can create the necessary parallel algorithms to realize a parallel adaptive numerical PDE solve.

We will provide theoretical and programming exercises for self study.