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Finite Element Method User Guide—Wolfram Language Documentation

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Learning & Support Get Started More Learning Grow Your Skills Tech Support Company Work with Us Educational Programs for Adults Educational Programs for Youth Read Educational Resources Wolfram Initiatives Events Wolfram|Alpha Wolfram Cloud Your Account Search Navigation Menu Wolfram Language & System Documentation Center Wolfram Language Home Page » FEM DOCUMENTATION PACKAGE OVERVIEW Finite Element Method User Guide Solving Partial Differential Equations with Finite Elements Introduction Why Finite Elements? What Is Needed for a Finite Element Analysis The Scope of the Finite Element Method as Implemented in NDSolve Regions Classical Partial Differential Equations The Coefficient Form of Partial Differential Equations Poisson’s Equation with Dirichlet Conditions Partial Differential Equations and Boundary Conditions Poisson’s Equation with Neumann Values Poisson’s Equation with a Periodic Boundary Condition Partial Differential Equations with Variable Coefficients Partial Differential Equations with Nonlinear Coefficients Partial Differential Equations with Nonlinear Variable Coefficients Partial Differential Equations and Nonlinear Boundary Conditions Heat Equation Wave Equation Formal Partial Differential Equations Systems of Partial Differential Equations The Coefficient Form of Systems of Partial Differential Equations One-Dimensional Coupled Partial Differential Equation Structural Mechanics Fluid Flow Element Mesh Generation Introduction Passing an ElementMesh to NDSolve Passing Options for the ElementMesh Creation to NDSolve via MeshOptions Comparing ElementMesh and MeshRegion Approximation of Regions with ElementMesh Manual Mesh Creation Line Meshes Triangle Meshes Quad Meshes Mixed Element Type Meshes in 2D Boundary Meshes in 2D Tetrahedron Meshes Hexahedron Meshes Mixed Element Type Meshes in 3D Boundary Meshes in 3D Region Approximation Quality Element Mesh Quality Visualize Low-Quality Elements Numerical Regions Element Meshes with Subregions Markers Element Meshes in Other Functions Region Membership Tests Interpolation Element Mesh Visualization Wireframes Issues Visualizing Deformations Converting ElementMesh Finite Element Method Usage Tips Introduction Monitoring the Solution Progress of Nonlinear Stationary Partial Differential Equations Monitoring Progress of Time Integration of Transient Partial Differential Equations Solving Memory-Intensive PDEs Extrapolation of Solution Domains Overshoot/Undershoot Issue for Discontinuous Coefficients Stabilization of Convection-Dominated Equations Stabilization of Convection-Dominated Time-Dependent Equations The Coefficient Form of a PDE Formal Partial Differential Equations NeumannValue and Formal Partial Differential Equations Efficient Evaluation of PDE Coefficients The Relation between NeumannValue and Boundary Derivatives Ordering of Dependent Variable Names Verifying Solutions Finite Element Programming Introduction Finite Element Data within NDSolve Passing Finite Element Options to NDSolve A Workflow Overview The Partial Differential Equation Problem Setup Stationary PDEs Initialization Stage Discretization Stage Solution Stage Post-Processing Nonlinear PDEs Top-Level Example Initialization Stage Discretization and Solution Stage Post-Processing The Linearization Process Solving the Linearized PDE Transient PDEs Transient PDE with Stationary Coefficients and Stationary Boundary Conditions—Introduction Transient PDE with Stationary Coefficients and Stationary Boundary Conditions Model Order Reduction of Transient PDEs with Stationary Coefficients and Stationary Boundary Conditions Transient PDEs with Transient Coefficients Transient PDEs with Nonlinear Transient Coefficients Transient PDEs with Integral Coefficients Coupled PDEs Deformation of a Beam under Load A Swinging Beam—Transient Coupled PDEs A Swinging and Dynamically Loaded Beam Large-Scale FEM Analysis NDSolve Options for Finite Elements Overview NDSolve Options The Method Option for Solution Stages How to Tell What Method Has Been Used What Triggers the Use of the Finite Element Method Finite Element Method Options for Stationary Partial Differential Equations InitializePDECoefficients Mesh Generation Options PDESolveOptions LinearSolver FindRootOptions Remaining "FiniteElement" Method Options Nonlinear Finite Element Method Verification Tests Stationary Tests 1D Single Equation Diffusion—FEM-NL-Stationary-1D-Single-Diffusion-0001 Diffusion—FEM-NL-Stationary-1D-Single-Diffusion-0002 Convection—FEM-NL-Stationary-1D-Single-Convection-0001 Convection—FEM-NL-Stationary-1D-Single-Convection-0002 Reaction—FEM-NL-Stationary-1D-Single-Reaction-0001 Reaction—FEM-NL-Stationary-1D-Single-Reaction-0002 Reaction—FEM-NL-Stationary-1D-Single-Reaction-0003 Reaction—FEM-NL-Stationary-1D-Single-Reaction-0004 Load—FEM-NL-Stationary-1D-Single-Load-0001 Load—FEM-NL-Stationary-1D-Single-Load-0002 Radiation BC—FEM-NL-Stationary-1D-Single-Radiation-0001 Missing test types 1D Systems of Equations Reaction—FEM-NL-Stationary-1D-System-Reaction-0001 2D Single Equation Diffusion—FEM-NL-Stationary-2D-Single-Diffusion-0001 Reaction—FEM-NL-Stationary-2D-Single-Reaction-0001 Transient Tests 1D Single Equation Diffusion—FEM-NL-Transient-1D-Single-Diffusion-0001 Diffusion—FEM-NL-Transient-1D-Single-Diffusion-0002 Diffusion—FEM-NL-Transient-1D-Single-Diffusion-0003 Convection—FEM-NL-Transient-1D-Single-Convection-0001 Convection—FEM-NL-Transient-1D-Single-Convection-0002 Reaction—FEM-NL-Transient-1D-Single-Reaction-0001 Reaction—FEM-NL-Transient-1D-Single-Reaction-0002 Load—FEM-NL-Transient-1D-Single-Load-0001 Load—FEM-NL-Transient-1D-Single-Load-0002 Reaction-Diffusion—FEM-NL-Transient-1D-Single-Reaction-Diffusion-0001 2D Single Equation Diffusion—FEM-NL-Stationary-2D-Single-Diffusion-0001 Test Result Inspection References ReferencePages/Symbols BoundaryConditionData BoundaryUnitNormal DeployBoundaryConditions DiscontinuousInterpolatingFunction DiscretizeBoundaryConditions DiscretizePDE DiscretizedBoundaryConditionData DiscretizedPDEData ElementMesh ElementMeshInterpolation ElementMeshProjection ElementMeshRegionProduct EvaluateOnElementMesh FEMMethodData FiniteElementData HexahedronElement InitializeBoundaryConditions InitializePDECoefficients InitializePDEMethodData LineElement NumericalRegion PDECoefficientData PDESolve PrismElement PointElement ProcessPDESolutions QuadElement TetrahedronElement ToBoundaryMesh ToElementMesh ToGradedMesh ToNumericalRegion TriangleElement Top

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