FEMDocumentation`
FEMDocumentation`

InitializePDECoefficients

InitializePDECoefficients[vd,sd,rules]

initializes the coefficients specified by rules in accordance with variable data vd and solution data sd to generate a PDECoefficientData object.

Details

  • The coefficients are assumed to come from a second-order system of PDEs in space dimensions of the form:
  • In InitializePDECoefficients[vd,sd,rules], the rules should be of the form name->coefficient. The possible coefficient names are:
  • "LoadCoefficients"{{f1},{f2},}f_(i) is a scalar
    "LoadDerivativeCoefficients"{{γ1},{γ2},}gamma_(i) is a vector of length n
    "DiffusionCoefficients"{{-c11,-c12,},{-c21,-c22,},}c_(ij) may be specified as scalars, diagonal vectors of length n, or n×n matrices
    "ConservativeConvectionCoefficients"{{-α11,-α12,},{-α21,-α22,},}alpha_(ij) is a vector of length n
    "ConvectionCoefficients"{{β11,β12,},{β21,β22,},}beta_(ij) is a vector of length n
    "ReactionCoefficients"{{a11,a12,},{a21,a22,},}a_(ij) is a scalar
    "DampingCoefficients"{{d11,d12,},{d21,d22,},}d_(ij) is a scalar
    "MassCoefficients"{{m11,m12,},{m21,m22,},}m_(ij) is a scalar
  • If a rule is not specified for any of these coefficient names, the coefficients of that type are all assumed to be 0.
  • Transient system of PDEs may be specified up through second order based on the form:
  • NDSolve reduces transient systems so that they are first order in time.
  • The coefficients can be functions of space, time, parameters, dependent variables and first order derivatives of dependent variables.
  • Variable data vd and solution data sd are corresponding lists of variables and values. Templates for vd and sd may be generated using NDSolve`VariableData and NDSolve`SolutionData, and components may be set using NDSolve`SetSolutionDataComponent.
  • InitializePDECoefficients verifies and optimizes the coefficients in accordance with variable data vd and solution data sd.
  • The "Space" component of vd and sd should be set to the spatial variables and the spatial mesh represented as a NumericalRegion object, respectively.
  • The "DependentVariables" component of vd should be a list of dependent variable name symbols without arguments.
  • For time-dependent problems, the "Time" component of vd and sd should be set to the temporal variable and the initial time, respectively.
  • For nonlinear problems, the "DependentVariables" component of sd should be set to the initial seedings for the dependent variables.
  • For parametric problems, the "Parameters" component of vd and sd should be set to the parametric variables and the initial parametric values, respectively.
  • InitializePDECoefficients has the following options:
  • "VerificationData"Automaticspecify PDE coefficient verification data
  • Options given to InitializePDECoefficients can be given to NDSolve by specifying "InitializePDECoefficientsOptions". »

Examples

open allclose all

Basic Examples  (2)

Load the finite element package:

Set up a NumericalRegion:

Set up variable and solution data:

Convert a Laplace equation into coefficients:

Convert a Poisson equation into coefficients:

Scope  (6)

Coefficients can be functions:

Coefficients can be interpolation functions:

Coefficients can be nonlinear:

If the input for "DiffusionCoefficient" is a vector, it is interpreted as the diagonal of the requested input matrix:

If the input for "DiffusionCoefficient" is a scalar, the scalar is put in all places of the diagonal of the requested matrix:

Initialize a system of PDEs:

Options  (1)

"VerificationData"  (1)

The PDE coefficients that are given to InitializePDECoefficients are verified so that they evaluate to numeric input:

The verification is done by finding a test coordinate that lies within the NumericalRegion. It may, however, happen that the automatically chosen test coordinate or test ElementMarker are at a singularity:

Overwriting the automatic finding of the verification data can be done with the "VerificationData" option:

Additionally, the automatic ElementMarker can be changed:

For nonlinear problems, the verification data for the dependents and the derivative of the dependents can be modified with "DependentVariables" and "DependentVariablesDerivative", respectively:

Change the test value for the dependent variable to be 1 and not 0 from the initial condition:

Change the test value of the derivative of the dependent variable to the vector {1,1}. The components of the vector are the values of the derivative in each space direction. Currently, a default 0.1 is used:

Applications  (4)

Specify a Laplace equation over a disk with Dirichlet and Neumann conditions for and for :

Inspect the diffusion coefficients:

Discretize, solve, interpolate and visualize the solution:

The "LoadCoefficients" coefficient is used by NIntegrate to integrate over a region:

Use NIntegrate to verify the result:

A "ReactionCoefficients" of can be used to evaluate a function, for example, over a region:

Visualize the evaluated function:

A "ReactionCoefficients" of can be used to scale a function, for example, over a region by a factor of :

Visualize the scaled evaluated function:

Properties & Relations  (1)

Options given to InitializePDECoefficients can be given to NDSolve by specifying "InitializePDECoefficientOptions":

Specify "VerificationData" to pass a coordinate that is not at a singularity:

Possible Issues  (1)

Typically, NDSolve will detect if coefficients do not evaluate to numeric quantities with the correct dimensions and warn about that:

To verify that the coefficients given are adequate for each operator, the coefficients undergo a test evaluation using the initial point in time and a coordinate from the region. If the combination of the coefficient and the used values cancel out, NDSolve cannot detect if a coefficient has not been specified. NDSolve will then give a message that it was not able to compute the elements contributed by a specific operator:

In the preceding case, evaluating the load operator at t=0 and x=-π will let the load operator vanish, and NDSolve cannot detect that k has not been specified: