The finite element method is a numerical method to solve differential equations over arbitrary-shaped domains. The finite element method is implemented in NDSolve as a spacial discretization method, and the primary usage of the finite element method is through NDSolve. Furthermore, interfaces to low-level finite element functionality are provided.

NDSolve — numerically solve differential equations

NIntegrate — numerically integrate

NDEigensystem — numerically compute differential eigenvalues and eigenvectors

Mesh Generation

ToBoundaryMesh — convert various input to a boundary mesh

ToElementMesh — convert various input to a full mesh

ToGradedMesh — convert Line graphics primitives input to a graded mesh

ElementMeshRegionProduct — the Cartesian products of element meshes

ElementMesh — a mesh data structure

NumericalRegion — a symbolic and mesh data structure

PointElement  ▪  LineElement  ▪  TriangleElement  ▪  QuadElement  ▪  TetrahedronElement  ▪  PrismElement  ▪  HexahedronElement

Initialization

InitializePDECoefficients — initialize partial differential equation coefficients

InitializeBoundaryConditions — initialize boundary conditions

InitializePDEMethodData — initialize partial differential equation method data

PDECoefficientData  ▪  BoundaryConditionData  ▪  FEMMethodData

Discretization

DiscretizePDE — discretize initialized partial differential equations

DiscretizeBoundaryConditions — discretize initialized boundary conditions

DiscretizedPDEData  ▪  DiscretizedBoundaryConditionData  ▪  DirichletCondition  ▪  NeumannValue  ▪  PeriodicBoundaryCondition

Solution

DeployBoundaryConditions — deploy discretized boundary conditions into discretized partial differential equations

LinearSolve — solve linear systems of equations

PDESolve — solve linear and nonlinear systems of equations

Post Processing

ProcessPDESolutions — process solution data into InterpolatingFunction objects

ElementMeshInterpolation — creates an InterpolatingFunction from a solution over a mesh

EvaluateOnElementMesh — creates an interpolating function from evaluating a function over a mesh

DiscontinuousInterpolatingFunction — creates discontinuous interpolating function from an InterpolatingFunction