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The deal.II tutorial step-100: An ultraweak Discontinuous Petrov-Galerkin method for the indefinite Helmholtz equation

Authors: Marquis, Oreste; Maier, Matthias; Blais, Bruno;

The deal.II tutorial step-100: An ultraweak Discontinuous Petrov-Galerkin method for the indefinite Helmholtz equation

Abstract

This tutorial for the deal.II finite element library demonstrates the implementation of the Discontinuous Petrov-Galerkin (DPG) method to solve the indefinite Helmholtz equation, which arises in the study of time-harmonic wave phenomena such as acoustic and electromagnetic propagation. Building on the well-posed Helmholtz problem of step-7, the program targets the more challenging high-frequency regime in which the indefiniteness of the operator causes classical Krylov methods and most blackbox preconditioners to break down. The DPG method is formulated as a residual minimization in a broken, enriched test space and always yields a Hermitian positive definite global system. This allows the use of a conjugate gradient solver in place of a direct solver or GMRES, substantially reducing memory consumption and enabling simulations at higher frequencies and on larger domains. The tutorial implements the ultraweak formulation of the linear acoustic equations on a 2D square domain, discretizing the full exact sequence of energy spaces using continuous Lagrange, Raviart-Thomas, and discontinuous Lagrange elements. The implementation shares many structural similarities with the hybridizable discontinuous Galerkin (HDG) approach of step-51: trace unknowns on the mesh skeleton couple neighboring cells, and static condensation via a Schur-complement procedure eliminates the interior unknowns to obtain a global system posed only on the skeleton degrees of freedom. Dirichlet, Neumann, and Robin boundary conditions are all demonstrated, with the Robin condition implemented following the approach of Gopalakrishnan and Schöberl in a manner that preserves the positive definiteness of the global system. The implementation is validated against an analytical plane-wave solution. Complete reference Compiling and running To generate a makefile for this code using CMake, create a build directory to your liking and type the following command into the terminal from the build directory: cmake /path/to/step-100 -DDEAL_II_DIR=/path/to/deal.II-9.8 To compile the program, call: make

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