pdejmps

(Not recommended) Error estimates for adaptation

pdejmps is not recommended. Use meshes represented as FEMesh objects instead of [p,e,t] meshes. For more information, see Version History.

Syntax

errf = pdejmps(p,t,c,a,f,u,alpha,beta,m)

Description

errf = pdejmps(p,t,c,a,f,u,alpha,beta,m) calculates the error indication function used for mesh adaptation. The columns of errf correspond to triangles, and the rows correspond to the equations in the PDE system.

The function computes the error indicator E(K) for each triangle K as

$$E\left(K\right)=\alpha {\Vert h^m\left(f-au\right)\Vert }_K+\beta {\left(\frac{1}{2}{\displaystyle \sum _{\tau \in \partial K}{h}_{\tau }^{2m}{[n_{\tau }\cdot (c\nabla u_h)]}^2}\right)}^{1/2}$$

where $$n_{\tau }$$ is the unit normal of edge $$\tau$$ and the braced term is the jump in flux across the element edge. Here, α and β are weight indices, and m is an order parameter. The norm is an L₂ norm computed over the element K.

Examples

Error Indication Function

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. Use the original coarser mesh and the refined mesh, and calculate the error indication function in both cases.

Generate and plot a mesh for the circle sector geometry.

[p,e,t] = initmesh('cirsg');
pdemesh(p,e,t)

Solve the Laplace equation.

u = assempde('cirsb',p,e,t,1,0,0);

Calculate the error indication function for each mesh triangle. Use the weight indices $$\alpha =0.15$$, $$\beta =0.15$$, and the order parameter m = 1.

alpha = 0.15;
beta = 0.15;
m = 1;
errf = pdejmps(p,t,1,0,0,u,alpha,beta,m);

Find the maximum value of the error indication function.

max(abs(errf))

ans = 
0.0306

Refine the original mesh and plot the result.

[p,e,t] = refinemesh('cirsg',p,e,t);
pdemesh(p,e,t)

Solve the same equation on the refined mesh, and calculate the error indication function for each mesh triangle. Use the same values for the weight indices and the order parameter.

u = assempde('cirsb',p,e,t,1,0,0);
errf = pdejmps(p,t,1,0,0,u,alpha,beta,m);

Find the maximum value of the error indication function.

max(abs(errf))

ans = 
0.0194

Solve the same equation using the adaptmesh function.

[u,p,e,t] = adaptmesh('cirsg','cirsb',1,0,0);

Number of triangles: 197
Number of triangles: 201
Number of triangles: 216
Number of triangles: 233
Number of triangles: 254
Number of triangles: 265
Number of triangles: 313
Number of triangles: 344
Number of triangles: 417
Number of triangles: 475
Number of triangles: 629

Maximum number of refinement passes obtained.

Plot the mesh.

pdemesh(p,e,t)

Calculate the error indication function for each mesh triangle.

errf = pdejmps(p,t,1,0,0,u,alpha,beta,m);

Find the maximum value of the error indication function.

max(abs(errf))

ans = 
0.0024

Input Arguments

p — Mesh node points

matrix

Mesh node points, specified as a 2-by-Np matrix of points (nodes), where Np is the number of nodes in the mesh. For details on mesh data representation, see initmesh.

Data Types: double

t — Mesh elements

4-by-Nt matrix

Mesh elements, specified as a 4-by-Nt matrix of triangles, where Nt is the number of triangles in the mesh. For details on mesh data representation, see initmesh.

Data Types: double

c — PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. c represents the c coefficient in the scalar PDE

$$-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

The coefficients c, a, and f can depend on the solution u if you use the nonlinear solver by setting the value of 'Nonlin' to 'on'. The coefficients cannot be functions of the time t.

Example: 'cosh(x+y.^2)'

Data Types: double | char | string | function_handle

a — PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. a represents the a coefficient in the scalar PDE

$$-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

The coefficients c, a, and f can depend on the solution u if you use the nonlinear solver by setting the value of 'Nonlin' to 'on'. The coefficients cannot be functions of the time t.

Example: 2*eye(3)

Data Types: double | char | string | function_handle

f — PDE coefficient

scalar | matrix | character vector | character array | string scalar | string vector | coefficient function

PDE coefficient, specified as a scalar, matrix, character vector, character array, string scalar, string vector, or coefficient function. f represents the f coefficient in the scalar PDE

$$-\nabla \cdot \left(c\nabla u\right)+au=f$$

or in the system of PDEs

$$-\nabla \cdot \left(c\otimes \nabla u\right)+au=f$$

The coefficients c, a, and f can depend on the solution u if you use the nonlinear solver by setting the value of 'Nonlin' to 'on'. The coefficients cannot be functions of the time t.

Example: char('sin(x)';'cos(y)';'tan(z)')

Data Types: double | char | string | function_handle

u — PDE solution

vector

PDE solution, specified as a vector.

• If the PDE is scalar, meaning that it has only one equation, then u is a column vector representing the solution u at each node in the mesh.

• If the PDE is a system of N > 1 equations, then u is a column vector with N*Np elements, where Np is the number of nodes in the mesh. The first Np elements of u represent the solution of equation 1, the next Np elements represent the solution of equation 2, and so on.

alpha — Weight index

number

Weight index, specified as number.

Data Types: double

beta — Weight index

number

Weight index, specified as a number.

Data Types: double

m — Order parameter

number

Order parameter, specified as number.

Data Types: double

Output Arguments

errf — Error indicator

matrix

Error indicator, returned as a matrix with the number of columns equal to the number of triangles t and the number of rows equal to the number of PDEs in the system.

• Each matrix row corresponds to an equation in the PDE system.

• Each column corresponds to a triangle.

Version History

Introduced before R2006a

R2016a: pdejmps is not recommended

pdejmps and [p,e,t] meshes are not recommended. Use meshes represented as FEMesh objects instead. There are no plans to remove pdejmps and [p,e,t] meshes.

Starting in R2016a, use the generateMesh function to create meshes as FEMesh objects. For details about these meshes, see Mesh Data.