EllipticForest

A Quadtree-Adaptive implementation of the Hierarchical Poincaré-Steklov (HPS) method for solving elliptic partial differential equations.

Features

• Solve elliptic PDEs on an adaptive mesh
• User-friendly implementation of the HPS method with VTK output
• An object-oriented code allows for user-expandability for additional patch solvers
• Ability to solve elliptic PDEs of the forms:
  • Laplace equation : $\Delta u(x,y) = 0$
  • Poisson equation : $\Delta u(x,y) = f(x,y)$
  • Helmholtz equation : $\Delta u(x,y) + \lambda u(x,y) = f(x,y)$
  • Variable coefficient elliptic equation : $\alpha(x,y) \nabla \cdot \left[\beta(x,y) \nabla u(x,y)\right] + \lambda(x,y) u(x,y) = f(x,y)$

Usage

Here is a minimum working example of using the HPS stages to solve an elliptic problem:

```C++

include

include

int main(int argc, char** argv) {

// ====================================================
// Initialize app and MPI
// ====================================================
EllipticForest::EllipticForestApp app(&argc, &argv);
EllipticForest::MPI::MPIObject mpi(MPI_COMM_WORLD);

// ====================================================
// Setup options
// ====================================================
bool cache_operators = false;
bool homogeneous_rhs = false;
bool vtk_flag = true;
double threshold = 1.2;
int n_solves = 1;
int min_level = 0;
int max_level = 7;
double x_lower = -10.0;
double x_upper = 10.0;
double y_lower = -10.0;
double y_upper = 10.0;
int nx = 16;
int ny = 16;

// ====================================================
// Create grid and patch prototypes
// ====================================================
EllipticForest::FiniteVolumeGrid grid(MPI_COMM_WORLD, nx, x_lower, x_upper, ny, y_lower, y_upper);
EllipticForest::FiniteVolumePatch root_patch(MPI_COMM_WORLD, grid);

// ====================================================
// Create node factory and mesh
// ====================================================
EllipticForest::FiniteVolumeNodeFactory node_factory(MPI_COMM_WORLD);
EllipticForest::Mesh<EllipticForest::FiniteVolumePatch> mesh{};
mesh.refineByFunction(
    [&](double x, double y){
        double f = -(sin(x) + sin(y));
        return fabs(f) > threshold;
    },
    threshold,
    min_level,
    max_level,
    root_patch,
    node_factory
);

// ====================================================
// Create patch solver
// ====================================================
EllipticForest::FiniteVolumeSolver solver{};
solver.solver_type = EllipticForest::FiniteVolumeSolverType::FISHPACK90;
solver.alpha_function = alphaFunction;
solver.beta_function = betaFunction;
solver.lambda_function = lambdaFunction;

// ====================================================
// Create and run HPS solver
// ====================================================
// 1. Create the HPSAlgorithm instance
EllipticForest::HPSAlgorithm<EllipticForest::FiniteVolumeGrid, EllipticForest::FiniteVolumeSolver, EllipticForest::FiniteVolumePatch, double> HPS(MPI_COMM_WORLD, mesh, solver);

// 2. Call the setup stage
HPS.setupStage();

// 3. Call the build stage
HPS.buildStage();

// 4. Call the upwards stage; provide a callback to set load data on leaf patches
HPS.upwardsStage([&](double x, double y){
    return -(sin(x) + sin(y));
});

// 5. Call the solve stage; provide a callback to set physical boundary Dirichlet data on root patch
HPS.solveStage([&](int side, double x, double y, double* a, double* b){
    *a = 1.0;
    *b = 0.0;
    return sin(x) + sin(y);
});

// All clean up is done in destructors
return EXIT_SUCCESS;

} ```

Running the above (which is a selection from examples/elliptic-single) outputs Unstructured Mesh VTK files which can be visualized with ParaView, VisIt, etc.:

Configuring, Building, and Installing

EllipticForest relies on several external packages for the mesh adaptivity, linear algebra, and parallelism:

Required: - MPI : Message-passing interface for distributed memory parallelism - BLAS and LAPACK : Basic Linear Algebra Subprograms and Linear Algebra PACKage - FISHPACK90 (GitHub) : A FORTRAN90 code for solving elliptic PDEs on structured grids (leaf patches). - p4est (GitHub) : Library for managing a collection of quadtrees or octrees for adaptive meshes. - petsc (GitLab) : Scalable linear algebra package for solving PDEs on distributed meshes.

Optional: - matplotlibcpp (GitHub) : A C++ wrapper for some matplotlib functionality.

The only packages the user is required to have installed are MPI, BLAS, and LAPACK; EllipticForest will attempt to build FISHPACK90, p4est, and petsc internally if not provided externally. Because petsc is not built with CMake, it is built with CMake's ExternalProject which adds a lot of extra compilation and additional steps. Ideally, the user will install petsc themselves and provide the path to it as detailed below. Instructions for installing petsc can be found here and can be done with apt install, Homebrew (Mac), and Spack.

The user may also provide paths to already installed versions of each of these packages and EllipticForest will use them accordingly.

EllipticForest uses CMake as the build system.

Configuration Examples

Minimum Configuration

```bash

cmake -S . -B build -DCMAKECXXCOMPILER=mpicxx -DCMAKECCOMPILER=mpicc -DMPIPATH=${PATHTO_MPI} ```

Pre-Installed Packages

```bash

cmake -S . -B build -DCMAKECXXCOMPILER=mpicxx -DCMAKECCOMPILER=mpicc -DMPIPATH=${PATHTOMPI} -DFISHPACK90PATH=${PATHTOFISHPACK90} -DP4ESTPATH=${PATHTOP4EST} -DPETSCPATH=${PATHTOPETSC} ```

matplotlibcpp Plotting Features

```bash

cmake -S . -B build -DCMAKECXXCOMPILER=mpicxx -DCMAKECCOMPILER=mpicc -DMPIPATH=${PATHTOMPI} -DWITHMATPLOTLIBCPP=true -DPYTHONENVPATH=${PYTHONENVPATH} -DPYTHONVERSION=${PYTHONVERSION} ```

Building Examples

Minimum Building

```bash

make ```

Testing Interface

EllipticForest uses GoogleTest and CTest for unit testing.

```bash

make test ```

Installation Examples

```bash

make install ```

Examples

There are some additional examples found in the examples directory. See the README therein for an overview of examples.

References

[1] S. Balay, S. Abhyankar, M. F. Adams, S. Benson, J. Brown, P. Brune, K. Buschelman, E. Constantinescu, L. Dalcin, A. Dener, V. Eijkhout, J. Faibussowitsch, W. D. Gropp, V. Hapla, T. Isaac, P. Jolivet, D. Karpeev, D. Kaushik, M. G. Knepley, F. Kong, S. Kruger, D. A. May, L. C. McInnes, R. T. Mills, L. Mitchell, T. Munson, J. E. Roman, K. Rupp, P. Sanan, J. Sarich, B. F. Smith, S. Zampini, H. Zhang, H. Zhang, and J. Zhang. PETSc/TAO users manual. Technical Report ANL-21/39 - Revision 3.20, Argonne National Laboratory, 2023.

[2] C. Burstedde, L. C. Wilcox, and O. Ghattas. p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM Journal on Scientific Computing, 33(3):1103–1133, 2011.

[3] A. Gillman and P.-G. Martinsson. A direct solver with O(N) complexity for variable coefficient ellip- tic PDEs discretized via a high-order composite spectral collocation method. SIAM J. Sci. Comput., 36(4):A2023–A2046, 2014.

[4] P. Martinsson. The hierarchical Poincar ́e-Steklov (HPS) solver for elliptic PDEs: A tutorial. arXiv preprint arXiv:1506.01308, 2015.

[5] P.-G. Martinsson. Fast direct solvers for elliptic PDEs. SIAM, 2019.