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:

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:

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 in combination with a "LoadCoefficients" can be used to scale a function, for example, over a region by a factor of :

Visualize the scaled evaluated function:

A "ReactionCoefficients" in combination with a "LoadDerivativeCoefficients" can be used to interpolate the derivative of a function over a region:

Visualize the difference between the numerical approximation and the exact derivative:

Specify an equation with mixed spatial and temporal derivatives in the domain :

Specify Dirichlet conditions , and initial conditions :

Set up the variable and solution data:

Initialize the coefficients:

Initialize the boundary conditions:

Initialize the method data:

Discretize the PDE coefficients. In this case, all PDE coefficients are time independent:

Inspect the stiffness matrix:

Inspect the damping matrix:

Discretize the boundary conditions:

Write a function to compute the residual of the PDE:

Evaluate the initial conditions:

Set up a function for the sparsity patterns of the system matrices:

Compute the sparsity patterns:

Solve the equation:

Create a unified interpolating function:

For comparison, compute a solution with the tensor product grid method:

Plot the difference between the TPG and FEM methods:

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

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

Convert a Poisson equation into coefficients:

Use GetInactivePDE to create an inactive version of the input:

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: