# FEM Problems

Below are the definitions of the types which specify problems. Some general notes are:

(t,x) vs (t,x,y): Mathematically one normally specifies equations in 2D as $f(t,x,y)$. However, in this code we use

`x`

as a vector. Thus you can think of $x$=`x[:,1]`

and $y$=`x[:,2]`

. Thus input equations are of the form`f(x,t)`

no matter the dimension. If time is not included in the problem (for example, a Poisson equation problem), then we use`f(x)`

. An example is the equation $u(x,y)= sin(2πx)cos(2πy)/(8π^2)$ would be specified as`sol(x) = sin(2π.*x[:,1]).*cos(2π.*x[:,2])/(8π*π)`

.Linearity: If the equation has a linear term, they are specified with functions

`f(t,x)`

. If it is nonlinear, it is specified with functions`f(t,x,u)`

. The boundary conditions are always`(t,x)`

Stochastic: By default the equation is deterministic. For each equation, one can specify a σ term which adds a stochastic $σ(t,x,u)dW_t$ term to the equation (or with $σ(t,x)dW_t$ if linear, must match

`f`

). $dW_t$ corresponds to the type of noise which is chosen. By default this is space-time Gaussian white noise.

## Poisson Equation Problem

Wraps the data that defines a 2D linear Poisson equation problem:

with bounday conditions `gD`

on the Dirichlet boundary and gN on the Neumann boundary. Linearity is determined by whether the forcing function `f`

is a function of one variable `(x)`

or two `(u,x)`

(with `x=[:,1]`

and `y=[:,2]`

).

If the keyword `σ`

is given, then this wraps the data that defines a 2D stochastic heat equation

### Constructors

`PoissonProblem(f,analytic,Du,mesh)`

: Defines the Dirichlet problem with analytical solution `analytic`

, solution gradient `Du = [u_x,u_y]`

, and forcing function `f`

`PoissonProblem(u0,f,mesh)`

: Defines the problem with initial value `u0`

(as a function) and f. If your initial data is a vector, wrap it as `u0(x) = vector`

.

Note: If all functions are of `(x)`

, then the program assumes it's linear. Write your functions using the math to program syntax translation: $x$ `= x[:,1]`

and $y$ `= x[:,2]`

. Use `f=f(u,x)`

and `σ=σ(u,x)`

(if specified) for nonlinear problems (with the boundary conditions still (x)). Systems of equations can be specified with `u_i = u[:,i]`

as the ith variable. See the example problems for more help.

### Keyword Arguments

`gD`

= Dirichlet boundary function`gN`

= Neumann boundary function`σ`

= The function which multiplies the noise $dW$. By default`σ=0`

.`noisetype`

= A string which specifies the type of noise to be generated. By default`noisetype=:White`

for Gaussian Spacetime White Noise.`numvars`

= The number of variables in the Poisson system. Automatically calculated in many cases.`D`

= Vector of diffusion coefficients. Defaults is`D=ones(1,numvars)`

.

## Heat Equation Problem

Wraps the data that defines a 2D heat equation problem:

with bounday conditions `gD`

on the Dirichlet boundary and gN on the Neumann boundary. Linearity is determined by whether the forcing function `f`

is a function of two variables `(t,x)`

or three `(t,x,u)`

(with `x=[:,1]`

and `y=[:,2]`

).

If the keyword `σ`

is given, then this wraps the data that defines a 2D stochastic heat equation.

### Constructors

`HeatProblem(analytic,Du,f,mesh)`

: Defines the Dirichlet problem with solution`analytic`

, solution gradient`Du = [u_x,u_y]`

, and the forcing function`f`

.`HeatProblem(u0,f,mesh)`

: Defines the problem with initial value`u0`

(as a function) and`f`

. If your initial data is a vector, wrap it as`u0(x) = vector`

.

Note: If all functions are of `(t,x)`

, then the program assumes it's linear. Write your functions using the math to program syntax translation: $x$ `= x[:,1]`

and $y$ `= x[:,2]`

. Use `f=f(t,x,u)`

and `σ=σ(t,x,u)`

(if specified) for nonlinear problems (with the boundary conditions still (t,x)). Systems of equations can be specified with `u_i = u[:,i]`

as the ith variable. See the example problems for more help.

### Keyword Arguments

`gD`

= Dirichlet boundary function`gN`

= Neumann boundary function`σ`

= The function which multiplies the noise dW. By default`σ=0`

.`noisetype`

= A string which specifies the type of noise to be generated. By default`noisetype=:White`

for Gaussian Spacetime White Noise.`numvars`

= Number of variables in the system. Automatically calculated from u0 in most cases.`D`

= Array which defines the diffusion coefficients. Default is`D=ones(1,numvars)`

.

## Example Problems

Examples problems can be found in DiffEqProblemLibrary.jl.

To use a sample problem, you need to do:

```
# Pkg.add("DiffEqProblemLibrary")
using DiffEqProblemLibrary
```

### Poisson Equation

Nonlinear Poisson equation with $f(u)=1-u/2$ and $f(v)=.5u-v$ and initial condition homogenous 1/2. Corresponds to the steady state of a humogenous reaction-diffusion equation with the same $f$.

`DiffEqProblemLibrary.prob_poisson_noisywave`

— Constant.Problem with deterministic solution: $u(x,y)= \sin(2πx)\cos(2πy)/(8π^2)$ and additive noise $σ(x,y)=5$

Nonlinear Poisson equation with $f(u)=1-u/2$ and $f(v)=1-v$ and initial condition homogenous 1/2. Corresponds to the steady state of a humogenous reaction-diffusion equation with the same $f$.

`DiffEqProblemLibrary.prob_poisson_wave`

— Constant.Problem defined by the solution: $u(x,y)= \sin(2πx)\cos(2πy)/(8π^2)$

`DiffEqProblemLibrary.prob_poisson_birthdeath`

— Constant.Nonlinear Poisson equation with $f(u)=1-u/2$. Corresponds to the steady state of a humogenous reaction-diffusion equation with the same $f$.

### Heat Equation

Homogenous reaction-diffusion which starts at 1/2 and solves the system $f(u)=1-u/2$ and $f(v)=1-v$

Homogenous reaction-diffusion which starts with 1/2 and solves the system $f(u)=1-u/2$ and $f(v)=.5u-v$

`DiffEqProblemLibrary.prob_femheat_diffuse`

— Constant.Example problem defined by the solution:

This is a Gaussian centered at $(\frac{1}{2},\frac{1}{2})$ which diffuses over time.

Homogenous stochastic reaction-diffusion problem which starts with 0 and solves with $f(u)=1-u/2$ with noise $σ(u)=10u^2$

`DiffEqProblemLibrary.prob_femheat_moving`

— Constant.Example problem defined by the solution:

This will have a mound which moves across the screen. Good animation test.

`DiffEqProblemLibrary.prob_femheat_pure`

— Constant.Example problem which starts with a Dirac δ cenetered at (0.5,0.5) and solves with $f=gD=0$. This gives the Green's function solution.

`DiffEqProblemLibrary.prob_femheat_birthdeath`

— Constant.Homogenous reaction-diffusion problem which starts with 0 and solves with $f(u)=1-u/2$