GPU-Accelerated Discontinuous Galerkin Methods on Polytopic Meshes

Discontinuous Galerkin (dG) methods are powerful tools for solving complex partial differential equations (PDEs), in particular those involving irregular geometries. Their key advantages include handling complex geometries and the ability to resolve localised features efficiently with adaptable meshes and/or variable local polynomial degrees. However, matrix assembly in dG implementations presents a challenge due to the use of physical frame basis functions and expensive quadratures over polytopic elements. This thesis addresses the challenge of efficient matrix assembly for arbitrary order quadrature rules on general polytopic meshes in the context of dG methods. The focus is on unstructured meshes with highly general polytopic elements, aiming for broad applicability. The approach utilizes a novel CUDA implementation targeting the hp-version interior penalty dG (IP-dG) method for equations with non-negative characteristic form. The core contribution lies in leveraging GPU parallel processing to overcome the computational challenges associated with arbitrary order quadrature methods on any polytopic domain. To achieve this, well-established quadrature rules are applied on subdivided simplices within the polytopic element, prioritizing method versatility and enabling efficient assembly for complex shapes. This thesis contributes to the field of GPU-accelerated dG methods by presenting novel and efficient assembly algorithms for problems with non-negative characteristic form. The presented methods offer broad applicability, efficient handling of complex shapes and good performance on both single and multi-GPU architectures.