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ideal.II: a Galerkin Space-Time Extension to the Finite Element Library deal.II

Jan Philipp Thiele

TL;DR

ideal.II extends deal.II to support space-time tensor-product finite elements, enabling all-at-once and slab-based formulations while preserving the library-like workflow. It provides a cohesive set of wrappers, data structures, and classes (e.g., DG_FiniteElement, FEValues, TimeIteratorCollection) that integrate with deal.II without reinventing core design, and supports arbitrary numbers of temporal intervals per slab to interpolate between time-stepping and all-at-once approaches. Through numerical experiments on the heat equation and Navier–Stokes flow, the framework demonstrates robust convergence behavior and compatibility with existing solvers, while enabling adjoint-oriented extensibility and dynamic triangulation in the future. The work delivers a practical, open-source toolkit for easy implementation of space-time PDEs with tensor-product discretizations and lays the groundwork for adaptive space-time refinement and advanced error controls in engineering applications.

Abstract

The C++ library deal.II provides classes and functions to solve stationary problems with finite elements on one- to threedimensional domains. It also supports the typical way to solve time-dependent problems using time-stepping schemes, either with an implementation by hand or through the use of external libraries like SUNDIALS. A different approach is the usage of finite elements in time as well, which results in space-time finite element schemes. The library ideal.II (short for instationary deal.II) aims to extend deal.II to simplify implementations of the second approach.

ideal.II: a Galerkin Space-Time Extension to the Finite Element Library deal.II

TL;DR

ideal.II extends deal.II to support space-time tensor-product finite elements, enabling all-at-once and slab-based formulations while preserving the library-like workflow. It provides a cohesive set of wrappers, data structures, and classes (e.g., DG_FiniteElement, FEValues, TimeIteratorCollection) that integrate with deal.II without reinventing core design, and supports arbitrary numbers of temporal intervals per slab to interpolate between time-stepping and all-at-once approaches. Through numerical experiments on the heat equation and Navier–Stokes flow, the framework demonstrates robust convergence behavior and compatibility with existing solvers, while enabling adjoint-oriented extensibility and dynamic triangulation in the future. The work delivers a practical, open-source toolkit for easy implementation of space-time PDEs with tensor-product discretizations and lays the groundwork for adaptive space-time refinement and advanced error controls in engineering applications.

Abstract

The C++ library deal.II provides classes and functions to solve stationary problems with finite elements on one- to threedimensional domains. It also supports the typical way to solve time-dependent problems using time-stepping schemes, either with an implementation by hand or through the use of external libraries like SUNDIALS. A different approach is the usage of finite elements in time as well, which results in space-time finite element schemes. The library ideal.II (short for instationary deal.II) aims to extend deal.II to simplify implementations of the second approach.
Paper Structure (45 sections, 1 theorem, 32 equations, 8 figures, 3 tables)

This paper contains 45 sections, 1 theorem, 32 equations, 8 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Space-time tensor-product finite elements
  3. Basic notation
  4. Bochner spaces
  5. Derivation of space-time weak formulations
  6. Discretization
  7. Resulting space-time shape functions and weak formulations
  8. Tensor-product slabs
  9. Extending a 1-to-3d finite element code to tensor product space-time
  10. Prerequisites
  11. Data structures
  12. Spacetime structures
  13. Slab structures
  14. Assembly
  15. Details of ideal.II
  16. ...and 30 more sections

Key Result

Theorem 2.2

For $W(I,V(\Omega))$ and $\mathbf{W}(I,\mathbf{V}(\Omega))$ as defined in def:Bochner the following continuous embeddings hold

Figures (8)

  • Figure 1: Assembly of $(\nabla u,\nabla\varphi)$ in the manual approach using deal.II
  • Figure 2: Assembly of $(\nabla u,\nabla\varphi)$ in the user-friendly approach using ideal.II
  • Figure 3: Results for uniform $h$-Refinement i. e. only in space
  • Figure 4: Results for uniform $k$-Refinement i. e. only in time
  • Figure 5: Results for uniform $kh$-Refinement i. e. in space and time simultaneously
  • ...and 3 more figures

Theorems & Definitions (5)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.6: Support for cG elements in time
  • Remark 2.7