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Introduction

GalerkinToolkit is a high-performance finite element toolbox fully implemented in the Julia programming language. Its mission is to provide general-purpose building blocks of finite element (FE) methods to solve a wide variety of partial differential equations (PDEs), using diverse numerical schemes, and across a range of computing platforms, from laptops to supercomputers.

The toolkit envisions a unified framework that addresses the needs of numerical analysts, domain scientists, and high-performance computing experts; offering a rich API with multiple levels of abstraction:

  • High-level API: Enables users to solve PDEs conveniently using mathematical abstractions instead of implementing low-level code by hand.
  • Low-level API: Grants direct access to underlying numerical quantities, allowing custom implementations of numerical schemes or advanced code optimizations not available through the high-level interface.

Features

GalerkinToolkit provides tools for:

  • Reading and partitioning computational meshes generated with external mesh generators.
  • Defining discrete interpolation spaces on triangulated manifolds, supporting both continuous and discontinuous interpolations.
  • Integrating functions, linear, and bilinear forms for scalar- and vector-valued equations, including single- and multi-field systems.
  • Supporting both simplex and hypercube face geometries, with high-order interpolation capabilities.
  • Discretizing a wide range of PDEs into systems of linear, nonlinear, and differential-algebraic equations.
  • Representing algebraic problems in formats compatible with external solvers.
  • Visualization and post-process of results.

Even though not currently available, these other features are being implemented, or are planned to be implemented in the near future:

  • Automatic differentiation for non-linear PDEs and gradient-based optimization methods.
  • Matrix-free bilinear forms.
  • H(div) and H(curl) interpolation spaces.
  • Distributed assembly.
  • Single- and multi-GPU support.

Novelties

GalerkinToolkit is certainly not the first FEM software project, but it introduces it introduces a novel design:

  • Unified high- and low-level APIs– It combines the vision of frameworks like FEniCS and libraries like Deal-ii in a single package and in a single programming language. It addresses the two-language problem of previous FE projects that consider a Python front-end for easiness of use and a C/C++ backend for performance. With GalerkinToolkit, you can use a concise high-level syntax or directly implement integration loops using low-level building blocks, depending on your needs.

  • Deep integration with the Julia ecosystem– GalerkinToolkit reuses existing Julia packages in numerous situations. For example:

  • The GalerkinToolkit Form Compiler (GTFC)– GTFC uses Julia’s metaprogramming capabilities to generate efficient code from high-level mathematical abstractions. Unlike other compilers like FFC or TSFC:

    • It does not rely on an external DSL like UFL. It considers a sub-set of the Julia programming language as alternative.
    • In consequence, it allows user-defined types in the weak form.
    • Moreover, it supports advanced use cases, such as coupling surface and volume arguments in multi-field weak forms both for continuous and discontinuous interpolations.

  • A major overhaul of Gridap– GalerkinToolkit began as a full reimplementation of the core ideas behind Gridap. While Gridap is based on lazily mapped arrays, GalerkinToolkit centers on form compilation and a low-level API designed to easily deal with quantities at integration points. These provide both a loop-free high-level API and manual control of integration loops.

Installation

GalerkinToolkit is a registered package in the official Julia package registry. As such, you can install GalerkinToolkit easily using the Julia package manager.

Open Julia and type ] to enter package mode.

julia> ]

Then, type

pkg> add GalerkinToolkit

This installs GalerkinToolkit and all its dependencies.

Press ctrl+C to go back to standard mode. Now, you can type

julia> import GalerkinToolkit as GT

to start using GalerkinToolkit.

You will need to have Julia installed in your system. See the official Julia installation instructions here.

This manual

It provides detailed explanations so that users build an understanding of the library. We start detailing the geometrical foundations of the library: computational meshes and domains. Then, we cover the construction of interpolation spaces and the numerical integration capabilities. Finally, we explain the post-processing layer including the visualization of results.