This project provides an efficient implementation of various adaptive mesh coarsening strategies in two dimensions. We coarsen meshes that are adaptively generated using the following refinement strategies.

For a use of the coarsening methods download the complete repository /ameshcoars with the test examples and run them on your computer with Matlab. To use it within your own examples you need the repository /refinement and /coarsening. Please see the following instructions for a correct use.

MATLAB

/example1: run test_moving_circle.m (refinement along a moving circle)

/example2: run example_P1_irr.m, exampleQ1_irr, and exampleP1Q1.m in the corresponding repositories. Vary the refinement method in exampleP1Q1.m (mesh refinement in the context of the adaptive finite element method for a quasi-stationary partial differential equation)

/example3: run example3.m (triangulation of a GIF)

/example4: run example4.m (local coarsening of a uniformly refined triangulation)

To be able to deploy this package within your framework you need the following data strucure of the mesh:

For triangles you need to define coordinates, elements3, and optionally boundary data dirichlet or neumann. Note, that for coarsening of TrefineR it is important to define the initial triangulation such that the smallest index is stored within an element at position one. In general, we number elements in a counterclockwise order.

Similarly, for quadrilaterals you need to define coordinates, elements4, and optionally boundary data dirichlet or neumann.

For meshes with hanging nodes an additional data vector named irregular is needed, where irregular(l,1) and irregular(l,2) are the starting and end point of the lth-irregular edge with hanging node stored in irregular(l,3). If there are no irregular edges, a predefinition irregular = zeros(0,3) is needed. Also note to clear the variables nG and nB before you start running your code as they are persistent variables!

We abbreviate the data structure as coordinates (C), elements3 (E3), elements4 (E4), irregular (I), dirichlet (D), and neumann (N). Furthermore, marked elements are stored with the corresponding index in the variable marked. N0 is the number of coordinates of the initial mesh.

The different mesh coarsening methods are then called by (remember that D and N are optional arguments)

[C,E3,I,D,N] = TcoarsenR(N0,C,E3,I,D,N,marked)

[C,E4,I,D,N] = QcoarsenR(N0,C,E4,I,D,N,marked)

[C,E3,D,N] = TcoarsenNVB(N0,C,E3,D,N,marked)

[C,E3,D,N] = TcoarsenRGB(N0,C,E3,D,N,marked)

[C,E3,D,N] = TcoarsenRG(N0,C,E3,D,N,marked)

[C,E4,D,N] = QcoarsenRB(N0,C,E4,D,N,marked)

[C,E3,E4,D,N] = Qcoarsen(N0,C,E3,E4,D,N,marked3,marked4)

The used functions are provided in the repository /example1 and /refinement:

addpath('../refinement')
addpath('../coarsening')

%% Define initial mesh
coordinates = [0,0;1,0;1,1;0,1;2,0;2,1];
elements = [3,1,2;1,3,4;2,6,3;6,2,5];
boundary = [1,2;2,5;5,6;6,3;3,4;4,1];
N0 = size(coordinates,1);
c_old = 0;

%% Refine uniformly
while 1
    marked = 1:size(elements,1);
    [coordinates,elements,boundary] ...
        = TrefineRGB(coordinates,elements,boundary,marked);
    if isempty(marked)|| (size(coordinates,1)>1e3)
        break
    end
end

% define discrete points (here a disc)
phi = -pi:pi/50:pi;
r = 0.2:1/50:0.4;
[r,phi] = meshgrid(r,phi);
s = r.*cos(phi);
t = r.*sin(phi);
points = [s(:)+1,t(:)+0.5];

%% Coarsen at discrete points
while 1
    mark3 = point2element(coordinates,elements,points);
    [coordinates,elements,boundary] = TcoarsenRGB(N0,coordinates,elements,boundary,mark3);
    if size(coordinates,1) == c_old
        break
    end
    c_old = size(coordinates,1);
end

% plot results
clf
patch('Faces',elements,'Vertices',coordinates,'Facecolor','none')
axis equal
axis off

triangular mesh:

patch('Faces',elements3,'Vertices',coordinates,'Facecolor','none')

quadrilateral mesh:

patch('Faces',elements4,'Vertices',coordinates,'Facecolor','none')

• Stefan A. Funken - Anja Schmidt Institute for Numerical Mathematics, Ulm University, Germany

If you use TcoarsenRGB in scientific work, please cite:

• Stefan A. Funken, and Anja Schmidt. "A Coarsening Algorithm on Adaptive Red-Green-Blue Refined Meshes", 2020

This project is licensed under the MIT License - see the LICENSE file for details