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

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