TensorGhost: Matrix-Free Ghost Penalty Evaluation via Tensor Product Factorization

Overview

TensorGhost is a high-performance implementation of ghost penalty stabilization for Cut Finite Element Methods (CutFEM), integrated within the deal.II finite element library.

This repository provides the source code and examples associated with the paper:

Matrix-Free Ghost Penalty Evaluation via Tensor Product Factorization by Michał Wichrowski.

The is matrix-free approach that exploits the tensor-product structure of high-order basis functions on Cartesian meshes. By factoring the ghost penalty operator into one-dimensional components, we significantly reduce computational complexity and memory overhead, enabling efficient high-order CutFEM simulations in 2D and 3D.

Key Features

• Matrix-Free CutFEM: Efficient evaluation of operators without explicit assembly of global matrices.
• Tensor Product Factorization: Implementation of ghost penalty stabilization using precomputed 1D mass and penalty matrices.
• High-Order Support: Optimized for high-order Lagrange finite elements (up to degree 8 and beyond).
• Scalable Design: Support for parallel distributed computations using MPI and vectorization.
• Integration with deal.II: Leverages the state-of-the-art MatrixFree framework.

Project Structure

• include/: Header files containing the core operators and generators.
  • ghost_penalty_operator.h: Linear operator for tensor-product ghost penalty.
  • matrix_free_operator.h: General matrix-free operator structure for CutFEM.
  • cut_cell_generator.h: Utilities for handling cut cell info.
• execs/: Primary executables and benchmarks.
  • matrix_free_global.cc: A global Laplace solver demonstrating the full CutFEM pipeline.
  • cell_comparison.cc: Benchmark comparing various evaluation strategies (FEEvaluation vs. GhostPenalty).
• source/: Library source files.
• doc/: Documentation and related manuscript files.

Requirements

• deal.II: Version 9.7 or later (the methods presented here are partially integrated into the 9.7 release).
• CMake: Version 3.10 or later.
• MPI: For parallel execution.
• C++17 compatible compiler.

Installation and Build

To configure and compile the project, point CMake to your deal.II installation:

mkdir build
cd build
cmake -Ddeal.II_DIR=/path/to/your/dealii/build/ ..
make -j4

Running Examples

After building, you can run the benchmarking and solver examples:

# Run the cell-wise benchmarking (2D/3D)
./cell_comparison_2d
./cell_comparison_3d

# Run the global matrix-free solver (2D/3D)
./matrix_free_global_2d
./matrix_free_global_3d

Citation

If you use this code in your research, please cite:

@article{wichrowski2025tensorghost,
  title={Matrix-Free Ghost Penalty Evaluation via Tensor Product Factorization},
  author={Wichrowski, Michał},
  year={2025},
  journal={TBD}
}

License

This project is licensed under the same terms as deal.II (LGPL v2.1 or later).