High Performance Solutions of Partial Differential Equations

This PhD course provides a focused introduction to high-performance computational methods for solving large-scale PDEs.

Topics include:

• Finite Element Method (FEM): Review of the theoretical foundations and practical aspects of discretizing PDEs into algebraic systems.

• deal.II Library: Hands-on implementation using the open-source deal.II library, designed for efficient and scalable FEM computations.

• Domain Decomposition Methods: Strategies for parallelization through subdomain partitioning to exploit modern high-performance architectures.

• Parallel Linear Algebra: Techniques for distributed linear solvers using PETSc and Trilinos, integrated with deal.II.

• Matrix-Free Geometric Multigrid Methods: Efficient solvers with minimal memory footprint and optimal computational scaling.

Through lectures and programming exercises, participants will gain the expertise to design and implement high-performance solvers for complex PDE-based models on modern computing platforms.

Prerequisites
Strong background in numerical analysis, linear algebra, and PDEs. Experience with C++ programming is highly recommended.

References

• A. Ern, J.-L. Guermond, Theory and Practice of Finite Elements, 2004

• J.J. Dongarra et al., Numerical Linear Algebra for High-Performance Computers, 1998

• A. Toselli, O.B. Widlund, Domain Decomposition Methods – Algorithms and Theory, 2005

• J.H. Bramble, X. Zhang, The Analysis of Multigrid Methods, 2000

Credits: 6 ECTS (30 hours)

Language: Italian/English

Exam: oral, including discussion of exercises or numerical projects