5-8
adaptmesh
Purpose Adaptive mesh generation and PDE solution.
Synopsis [u,p,e,t]=adaptmesh(g,b,c,a,f)
[u,p,e,t]=adaptmesh(g,b,c,a,f,'PropertyName',PropertyValue,)
Description [u,p,e,t]=adaptmesh(g,b,c,a,f,'PropertyName',PropertyValue,)
performs adaptive mesh generation and PDE solution. Optional arguments are given as property name/property value pairs.
The function produces a solution u to the elliptic scalar PDE problem –∇ ⋅ (c∇u) + au = f on Ω,
or the elliptic system PDE problem –∇ ⋅ (c ⊗ ∇u) + au = f on Ω
with the problem geometry and boundary conditions given by g and b. The mesh is described by the p, e, and t.
The solution u is represented as the solution vector u. Details on the representation of the solution vector can be found in the entry on assempde. The algorithm works by solving a sequence of PDE problems using refined triangular meshes. The first triangular mesh generation is obtained either as an optional argument to adaptmesh or by a call to initmesh without options.
The following generations of triangular meshes are obtained by solving the PDE problem, computing an error estimate, selecting a set of triangles based on the error estimate, and then finally refining these triangles. The solution to the PDE problem is then recomputed. The loop continues until no triangles are selected by the triangle selection method, or until the maximum number of triangles is attained, or until the maximum number of triangle generations has been generated.
gdescribes the decomposed geometry of the PDE problem. g can either be a Decomposed Geometry matrix or the name of a Geometry M-file. The formats of the Decomposed Geometry matrix and Geometry M-file are described in the entries on decsg and pdegeom, respectively.
bdescribes the boundary conditions of the PDE problem. b can be either a Boundary Condition matrix or the name of a Boundary M-file. The formats of the Boundary Condition matrix and Boundary M-file are described in the entries on assemb and pdebound, respectively.
adaptmesh
5-9 The adapted triangular mesh of the PDE problem is given by the mesh data p, e, and t. Details on the mesh data representation can be found in the entry on initmesh.
The coefficients c, a, and f of the PDE problem can be given in a wide variety of ways. In the context of adaptmesh the coefficients can depend on u if the nonlinear solver is enabled using the property nonlin. The coefficients cannot depend on t, the time. A complete listing of all options can be found in the entry on assempde.
The table below lists the property name/property value pairs, their default values, and descriptions of the properties.
Par is passed to the Tripick function. (The Tripick function is described below.) Normally it is used as tolerance of how well the solution fits the equation.
Property Property Default Description
Maxt positive integer inf Maximum number of new
triangles
Ngen positive integer 10 Maximum number of triangle
generations
Mesh p1, e1, t1 initmesh Initial mesh
Tripick MATLAB function pdeadworst Triangle selection method
Par numeric 0.5 Function parameter
Rmethod longest|regular longest Triangle refinement method
Nonlin on|off off Use nonlinear solver
Toln numeric 1e–4 Nonlinear tolerance
Init u0 0 Nonlinear initial value
Jac fixed|lumped|full fixed Nonlinear Jacobian
calculation
norm numeric|inf|energy inf Nonlinear residual norm
adaptmesh
5-10
No more than Ngen successive refinements are attempted. Refinement is also stopped when the number of triangles in the mesh exceeds Maxt.
p1, e1, and t1 are the input mesh data. This triangular mesh is used as starting mesh for the adaptive algorithm. Details on the mesh data representation can be found in the entry on initmesh. If no initial mesh is provided, the result of a call to initmesh with no options is used as initial mesh.
The triangle selection method, Tripick, is a user-definable triangle selection method. Given the error estimate computed by the function pdejmps, the triangle selection method selects the triangles to be refined in the next triangle generation. The function is called using the arguments p, t, cc, aa, ff, u, errf, and par. p and t represent the current generation of triangles, cc, aa, and ff are the current coefficients for the PDE problem, expanded to triangle
midpoints, u is the current solution, errf is the computed error estimate, and par, the function parameter, given to adaptmesh as optional argument. The matrices cc, aa, ff, and errf all have Nt columns, where Nt is the current number of triangles. The number of rows in cc, aa, and ff are exactly the same as the input arguments c, a, and f. errf has one row for each equation in the system. There are two standard triangle selection methods in the toolbox — pdeadworst and pdeadgsc. pdeadworst selects triangles where errf exceeds a fraction (default: 0.5) of the the worst value, and pdeadgsc selects triangles using a relative tolerance criterion.
The refinement method is either longest or regular. Details on the refinement method can be found in the entry on refinemesh.
The adaptive algorithm can also solve nonlinear PDE problems. For nonlinear PDE problems, the Nonlin parameter must be set to on. The nonlinear tolerance Toln, nonlinear initial value u0, nonlinear Jacobian calculation Jac, and nonlinear residual norm Norm are passed to the nonlinear solver
pdenonlin. See the entry on pdenonlin for details on the nonlinear solver.
adaptmesh
5-11 Examples Solve the Laplace equation over a circle sector, with Dirichlet boundary
conditions u = cos(2/3atan2(y,x)) along the arc, and u = 0 along the straight lines, and compare to the exact solution. We refine the triangles using the worst error criterion until we obtain a mesh with at least 500 triangles:
» [u,p,e,t]=adaptmesh('cirsg','cirsb',1,0,0,'maxt',500,...
'tripick','pdeadworst','ngen',inf);
» x=p(1,:); y=p(2,:);
» exact=((x.^2+y.^2).(1/3).*cos(2/3*atan2(y,x)))';
» max(abs(u–exact)) ans =
0.0058
» size(t,2) ans = 534
» pdemesh(p,e,t)
adaptmesh
5-12
The maximum absolute error is 0.0058, with 534 triangles. We test how many refinements we have to use with an uniform triangle net:
» [p,e,t]=initmesh('cirsg');
» [p,e,t]=refinemesh('cirsg',p,e,t);
» u=assempde('cirsb',p,e,t,1,0,0);
» x=p(1,:); y=p(2,:);
» exact=((x.^2+y.^2).^(1/3).*cos(2/3*atan2(y,x)))';
» max(abs(u–exact)) ans =
0.0085
» size(t,2) ans =
1640
» [p,e,t]=refinemesh('cirsg',p,e,t);
» u=assempde('cirsb',p,e,t,1,0,0);
» x=p(1,:); y=p(2,:);
» exact=((x.^2+y.^2).^(1/3).*cos(2/3*atan2(y,x)))';
» max(abs(u–exact)) ans =
0.0054
» size(t,2) ans =
6560
» pdemesh(p,e,t)
Thus, with uniform refinement, we need 6560 triangles to achieve better absolute error than what we achieved with the adaptive method. Note that the error is reduced only by 0.6 when the number of elements in quadrupled by the uniform refinement. For a problem with regular solution, we expect a O(h2) error, but this solution is singular since u ≈ r1/3 at the origin.
Diagnostics Upon termination, one of the following messages is displayed:
•Adaption completed (This means that the Tripick function returned zero triangles to refine.)
•Maximum number of triangles obtained
•Maximum number of refinement passes obtained
See Also initmesh, refinemesh, assempde, pdeadgsc, pdeadworst, pdejmps