MATLAB Partial Differential Equation Toolbox™ User's Guide

2064 Pages·2022·16.784 MB·English

by  The MathWorks, Inc.

Partial Differential Equation Toolbox™
User's Guide
R2022b
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Partial Differential Equation Toolbox™ User's Guide
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Revision History
August 1995 First printing New for Version 1.0
February 1996 Second printing Revised for Version 1.0.1
July 2002 Online only Revised for Version 1.0.4 (Release 13)
September 2002 Third printing Minor Revision for Version 1.0.4
June 2004 Online only Revised for Version 1.0.5 (Release 14)
October 2004 Online only Revised for Version 1.0.6 (Release 14SP1)
March 2005 Online only Revised for Version 1.0.6 (Release 14SP2)
August 2005 Fourth printing Minor Revision for Version 1.0.6
September 2005 Online only Revised for Version 1.0.7 (Release 14SP3)
March 2006 Online only Revised for Version 1.0.8 (Release 2006a)
March 2007 Online only Revised for Version 1.0.10 (Release 2007a)
September 2007 Online only Revised for Version 1.0.11 (Release 2007b)
March 2008 Online only Revised for Version 1.0.12 (Release 2008a)
October 2008 Online only Revised for Version 1.0.13 (Release 2008b)
March 2009 Online only Revised for Version 1.0.14 (Release 2009a)
September 2009 Online only Revised for Version 1.0.15 (Release 2009b)
March 2010 Online only Revised for Version 1.0.16 (Release 2010a)
September 2010 Online only Revised for Version 1.0.17 (Release 2010b)
April 2011 Online only Revised for Version 1.0.18 (Release 2011a)
September 2011 Online only Revised for Version 1.0.19 (Release 2011b)
March 2012 Online only Revised for Version 1.0.20 (Release 2012a)
September 2012 Online only Revised for Version 1.1 (Release 2012b)
March 2013 Online only Revised for Version 1.2 (Release 2013a)
September 2013 Online only Revised for Version 1.3 (Release 2013b)
March 2014 Online only Revised for Version 1.4 (Release 2014a)
October 2014 Online only Revised for Version 1.5 (Release 2014b)
March 2015 Online only Revised for Version 2.0 (Release 2015a)
September 2015 Online only Revised for Version 2.1 (Release 2015b)
March 2016 Online only Revised for Version 2.2 (Release 2016a)
September 2016 Online only Revised for Version 2.3 (Release 2016b)
March 2017 Online only Revised for Version 2.4 (Release 2017a)
September 2017 Online only Revised for Version 2.5 (Release 2017b)
March 2018 Online only Revised for Version 3.0 (Release 2018a)
September 2018 Online only Revised for Version 3.1 (Release 2018b)
March 2019 Online only Revised for Version 3.2 (Release 2019a)
September 2019 Online only Revised for Version 3.3 (Release 2019b)
March 2020 Online only Revised for Version 3.4 (Release 2020a)
September 2020 Online only Revised for Version 3.5 (Release 2020b)
March 2021 Online only Revised for Version 3.6 (Release 2021a)
September 2021 Online only Revised for Version 3.7 (Release 2021b)
March 2022 Online only Revised for Version 3.8 (Release 2022a)
September 2022 Online only Revised for Version 3.9 (Release 2022b)

Contents
Getting Started
1
Partial Differential Equation Toolbox Product Description . . . . . . . . . . . . 1-2
Key Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-2
Equations You Can Solve Using PDE Toolbox . . . . . . . . . . . . . . . . . . . . . . . 1-3
Solve 2-D PDEs Using the PDE Modeler App . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-5
Poisson’s Equation with Complex 2-D Geometry: PDE Modeler App . . . . . 1-7
Finite Element Method Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-11
Deflection Analysis of Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-14
Heat Transfer in Block with Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25
Setting Up Your PDE
2
Solve Problems Using PDEModel Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 2-2
Geometry and Mesh Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4
2-D Geometry Creation at Command Line . . . . . . . . . . . . . . . . . . . . . . . . . 2-16
Three Elements of Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16
Basic Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-16
Rectangle with Circular End Cap and Another Circular Excision . . . . . . . 2-17
Decomposed Geometry Data Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 2-20
Parametrized Function for 2-D Geometry Creation . . . . . . . . . . . . . . . . . 2-22
Required Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-22
Relation Between Parametrization and Region Labels . . . . . . . . . . . . . . . 2-22
Geometry Function for a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-23
Arc Length Calculations for a Geometry Function . . . . . . . . . . . . . . . . . . 2-24
Geometry Function Example with Subdomains and a Hole . . . . . . . . . . . 2-33
Nested Function for Geometry with Additional Parameters . . . . . . . . . . . 2-35
Geometry from polyshape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-39
STL File Import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-43
v
STEP File Import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-59
Geometry from Triangulated Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-63
3-D Geometry from a Finite Element Mesh . . . . . . . . . . . . . . . . . . . . . . . 2-63
2-D Multidomain Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-64
Geometry from alphaShape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-66
Cuboids, Cylinders, and Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-68
Sphere in Cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-75
3-D Multidomain Geometry from 2-D Geometry . . . . . . . . . . . . . . . . . . . . 2-79
Multidomain Geometry Reconstructed from Mesh . . . . . . . . . . . . . . . . . . 2-83
Put Equations in Divergence Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-87
Coefficient Matching for Divergence Form . . . . . . . . . . . . . . . . . . . . . . . 2-87
Boundary Conditions Can Affect the c Coefficient . . . . . . . . . . . . . . . . . . 2-88
Coefficient Conversion with Symbolic Math Toolbox . . . . . . . . . . . . . . . . 2-88
Some Equations Cannot Be Converted . . . . . . . . . . . . . . . . . . . . . . . . . . 2-89
f Coefficient for specifyCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-90
c Coefficient for specifyCoefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-92
Overview of the c Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-92
Definition of the c Tensor Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-92
Some c Vectors Can Be Short . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-94
Functional Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-104
m, d, or a Coefficient for specifyCoefficients . . . . . . . . . . . . . . . . . . . . . 2-107
Coefficients m, d, or a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-107
Short m, d, or a vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-107
Nonconstant m, d, or a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-108
View, Edit, and Delete PDE Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 2-111
View Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-111
Delete Existing Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-112
Change a Coefficient Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-113
Set Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-114
What Are Initial Conditions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-114
Constant Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-114
Nonconstant Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-114
Nodal Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-115
Nonlinear System with Cross-Coupling Between Components . . . . . . . 2-117
Set Initial Condition for Model with Fine Mesh Using Solution Obtained
with Coarser Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-121
View, Edit, and Delete Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 2-123
View Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-123
Delete Existing Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-124
Change an Initial Conditions Assignment . . . . . . . . . . . . . . . . . . . . . . . 2-125
vi Contents
No Boundary Conditions Between Subdomains . . . . . . . . . . . . . . . . . . . 2-126
Identify Boundary Labels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-128
Specify Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-129
Dirichlet Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-129
Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-130
Mixed Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-132
Nonconstant Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-132
Additional Arguments in Functions for Nonconstant Boundary Conditions
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-134
Solve PDEs with Constant Boundary Conditions . . . . . . . . . . . . . . . . . . 2-135
Specify Nonconstant Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 2-139
Geometry and Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-139
Scalar PDE Problem with Nonconstant Boundary Conditions . . . . . . . . 2-140
Anonymous Functions for Nonconstant Boundary Conditions . . . . . . . . 2-142
Additional Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-142
System of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-143
Specify Nonconstant PDE Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 2-146
Geometry and Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-146
Function for Nonconstant Coefficient f . . . . . . . . . . . . . . . . . . . . . . . . . 2-146
Anonymous Function for a PDE Coefficient . . . . . . . . . . . . . . . . . . . . . . 2-148
Additional Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-148
View, Edit, and Delete Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 2-150
View Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-150
Delete Existing Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 2-152
Change a Boundary Conditions Assignment . . . . . . . . . . . . . . . . . . . . . 2-152
Generate Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-154
Find Mesh Elements and Nodes by Location . . . . . . . . . . . . . . . . . . . . . . 2-162
Assess Quality of Mesh Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-168
Mesh Data as [p,e,t] Triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-172
Mesh Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-175
Solving PDEs
3
von Mises Effective Stress and Displacements: PDE Modeler App . . . . . . 3-3
Clamped Square Isotropic Plate with Uniform Pressure Load . . . . . . . . . 3-7
Deflection of Piezoelectric Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11
vii
Dynamics of Damped Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-21
Dynamic Analysis of Clamped Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-28
Reduced-Order Modeling Technique for Beam with Point Load . . . . . . . 3-33
Modal and Frequency Response Analysis for Single Part of Kinova Gen3
Robotic Arm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-40
Thermal Stress Analysis of Jet Engine Turbine Blade . . . . . . . . . . . . . . . 3-52
Finite Element Analysis of Electrostatically Actuated MEMS Device . . . 3-59
Deflection Analysis of Bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-74
Vibration of Square Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-85
Structural Dynamics of Tuning Fork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-90
Modal Superposition Method for Structural Dynamics Problem . . . . . . 3-99
Stress Concentration in Plate with Circular Hole . . . . . . . . . . . . . . . . . . 3-102
Thermal Deflection of Bimetallic Beam . . . . . . . . . . . . . . . . . . . . . . . . . . 3-110
Axisymmetric Thermal and Structural Analysis of Disc Brake . . . . . . . 3-119
Electrostatic Potential in Air-Filled Frame . . . . . . . . . . . . . . . . . . . . . . . 3-130
Electrostatic Potential in Air-Filled Frame: PDE Modeler App . . . . . . . 3-133
Electrostatic Analysis of Transformer Bushing Insulator . . . . . . . . . . . 3-135
Magnetic Flux Density in H-Shaped Magnet . . . . . . . . . . . . . . . . . . . . . . 3-141
Magnetic Flux Density in Electromagnet . . . . . . . . . . . . . . . . . . . . . . . . 3-146
Linear Elasticity Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-156
Summary of the Equations of Linear Elasticity . . . . . . . . . . . . . . . . . . . 3-156
3D Linear Elasticity Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-157
Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-159
Plane Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-160
Magnetic Field in Two-Pole Electric Motor . . . . . . . . . . . . . . . . . . . . . . . 3-161
Magnetic Field in Two-Pole Electric Motor: PDE Modeler App . . . . . . . 3-166
Helmholtz Equation on Disk with Square Hole . . . . . . . . . . . . . . . . . . . . 3-170
Electrostatics and Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-176
Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-176
Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-176
Magnetostatics with Permanent Magnets . . . . . . . . . . . . . . . . . . . . . . . 3-177
viii Contents
DC Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-178
Harmonic Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-179
Current Density Between Two Metallic Conductors . . . . . . . . . . . . . . . . 3-181
Skin Effect in Copper Wire with Circular Cross Section: PDE Modeler App
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-184
Current Density Between Two Metallic Conductors: PDE Modeler App
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-192
Heat Transfer Between Two Squares Made of Different Materials: PDE
Modeler App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-195
Temperature Distribution in Heat Sink . . . . . . . . . . . . . . . . . . . . . . . . . . 3-199
Create 2-D Geometry in PDE Modeler App . . . . . . . . . . . . . . . . . . . . . . 3-199
Extrude 2-D Geometry into 3-D Geometry of Heat Sink . . . . . . . . . . . . . 3-200
Perform Thermal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-203
Nonlinear Heat Transfer in Thin Plate . . . . . . . . . . . . . . . . . . . . . . . . . . 3-210
Poisson's Equation on Unit Disk: PDE Modeler App . . . . . . . . . . . . . . . 3-218
Poisson's Equation on Unit Disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-224
Scattering Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-232
Scattering Problem: PDE Modeler App . . . . . . . . . . . . . . . . . . . . . . . . . . 3-237
Minimal Surface Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-241
Minimal Surface Problem: PDE Modeler App . . . . . . . . . . . . . . . . . . . . . 3-247
Poisson's Equation with Point Source and Adaptive Mesh Refinement 3-249
Heat Transfer in Block with Cavity: PDE Modeler App . . . . . . . . . . . . . 3-254
Heat Transfer in Block with Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-258
Heat Transfer Problem with Temperature-Dependent Properties . . . . . 3-262
Heat Conduction in Multidomain Geometry with Nonuniform Heat Flux
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-270
Inhomogeneous Heat Equation on Square Domain . . . . . . . . . . . . . . . . 3-277
Heat Distribution in Circular Cylindrical Rod . . . . . . . . . . . . . . . . . . . . 3-281
Thermal Analysis of Disc Brake . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-287
Heat Distribution in Circular Cylindrical Rod: PDE Modeler App . . . . . 3-295
Wave Equation on Square Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-298
ix
Wave Equation on Square Domain: PDE Modeler App . . . . . . . . . . . . . . 3-302
Eigenvalues and Eigenmodes of L-Shaped Membrane . . . . . . . . . . . . . . 3-305
Eigenvalues and Eigenmodes of L-Shaped Membrane: PDE Modeler App
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-311
L-Shaped Membrane with Rounded Corner: PDE Modeler App . . . . . . 3-314
Eigenvalues and Eigenmodes of Square . . . . . . . . . . . . . . . . . . . . . . . . . 3-317
Eigenvalues and Eigenmodes of Square: PDE Modeler App . . . . . . . . . 3-322
Vibration of Circular Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-325
Solution and Gradient Plots with pdeplot and pdeplot3D . . . . . . . . . . . 3-329
2-D Solution and Gradient Plots with MATLAB Functions . . . . . . . . . . . 3-338
3-D Solution and Gradient Plots with MATLAB Functions . . . . . . . . . . . 3-344
Types of 3-D Solution Plots Available in MATLAB . . . . . . . . . . . . . . . . . 3-344
2-D Slices Through 3-D Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-344
Contour Slices Through 3-D Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 3-347
Plots of Gradients and Streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-351
Dimensions of Solutions, Gradients, and Fluxes . . . . . . . . . . . . . . . . . . 3-356
PDE Modeler App
4
Open the PDE Modeler App . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2
2-D Geometry Creation in PDE Modeler App . . . . . . . . . . . . . . . . . . . . . . . 4-3
Create Basic Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-3
Select Several Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4
Rotate Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4
Create Complex Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-4
Adjust Axes Limits and Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5
Create Geometry with Rounded Corners . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8
Specify Boundary Conditions in the PDE Modeler App . . . . . . . . . . . . . . 4-12
Specify Coefficients in PDE Modeler App . . . . . . . . . . . . . . . . . . . . . . . . . 4-14
Coefficients for Scalar PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-14
Coefficients for Systems of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-16
Coefficients That Depend on Time and Space . . . . . . . . . . . . . . . . . . . . . 4-18
Specify Mesh Parameters in the PDE Modeler App . . . . . . . . . . . . . . . . . 4-24
Adjust Solve Parameters in the PDE Modeler App . . . . . . . . . . . . . . . . . . 4-26
Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-26
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