Now let us look at another class of computational problems, the so-called advection-diffusion problem. In this problem, the differential equation in the so-called strong form is given by a formula
\( \beta_x\frac{\partial u}{\partial x }+\beta_y\frac{\partial u }{\partial y }-\epsilon \left(\frac{\partial^2 u }{\partial x^2 }+\frac{\partial^2 u }{\partial y^2 }\right)=f(x,y) \)
We are looking for a field of concentration, e.g. pollution \( \Omega\in (x,y) \rightarrow u(x,y) \in R \) in the area \( \Omega \). Value \( u(x,y) \) is the concentration of pollutants in a point \( (x,y) \).
The term \( \beta_x(x,y)\frac{\partial u }{\partial x }+\beta_y(x,y)\frac{\partial u }{\partial y } \) here means the advection, that is, the dispersion of pollutants by the wind.
The terms \( (\beta_x(x,y),\beta_y(x,y)) \) denote the components of the wind speed field, i.e. at a point \( (x,y) \) the wind blows in a direction parallel to the axis \( x \) with the speed \( \beta_x(x,y) \) in a direction parallel to the axis \( y \) with the speed \( \beta_y(x,y) \).
On the other hand, the term \( \epsilon \left(\frac{\partial^2 u }{\partial x^2 }+\frac{\partial^2 u }{\partial y^2 }\right) \) stands for diffusion, i.e. the distribution of pollutants by diffusion.
If I am sitting in a room with a smoker, even if there is no airflow, the smoke of the cigarette will reach my nose by a diffusion mechanism. The rate of this diffusion is responsible for the coefficient \( \epsilon \). If, additionally, the windows and doors were open in the room, cigarette smoke would also be pushed by wind blowing at a speed of e.g. \( (1,0) \) (if the window-door is in a plane parallel to the axis \( x \) and the airflow moves with speed 1 m/s).
On the right-hand side, the function \( f(x,y) \) stands for the source of contamination, that is, if in point \( (x,y) \) there is a chimney that emits pollution, then \( f(x,y)>0 \), if, however, in point \( (x,y) \) there is an aspirator of pollutants (recently such pollutants are sucked by the Chinese in the cities) then \( f(x,y)<0 \). Additionally \( f(x,y)=0 \) means that at point \( (x,y) \) no pollution occurs or diminishes.
Note that the diffusion term contains Laplasian and the advection term contains the dot product of the vector
\( \beta \) and wind speed vector. So we can rewrite our problem like this:
\( \beta(x,y) \cdot \nabla u(x,y) - \epsilon \Delta u(x,y) = f(x,y) \)
The weak formulation is obtained as follows. We integrate and multiply our equation by selected functions \( v(x,y) \) called test functions which we will use to average our equation in the area where these functions are defined

In a similar way to the bitmap projection problem, each selection of a test function \( v(x,y) \) for averaging our problem in the area where the test function is defined gives us one equation. Various test function selections \( v(x,y) \) give us different equations ( 1 ).
So our problem that we want to solve looks like this: We are looking for pollution concentration, functions \( \Omega \ni (x,y) \leftarrow u(x,y) \in R \) such that
\( \int_{\Omega} \beta_x(x,y) \frac{\partial u(x,y) }{\partial x } dxdy \int_{\Omega } \beta_y(x,y) \frac{\partial u(x,y) }{\partial y } dxdy \\ -\int_{\Omega } \frac{\partial^2 u(x,y) }{\partial x^2 } v(x,y) dxdy -\int_{\Omega } \frac{\partial^2 u(x,y) }{\partial y^2 } v(x,y) dxdy = \\ \int_{\Omega } f(x,y) v(x,y) dxdy \)
for any test functions \( \Omega \ni (x,y) \rightarrow v(x,y) \in R \) (of course our pollutant concentration and test function must be regular enough to be able to compute the integrals).
We denote our problem:
Find \( \Omega \ni (x,y) \rightarrow u(x,y) \in R \) such that
\( a(u,v)=l(v) \\ a(u,v) = \int_{\Omega} \beta_x(x,y) \frac{\partial u(x,y)}{\partial x} dxdy+ \int_{\Omega } \beta_y(x,y) \frac{\partial u(x,y)}{\partial y } dxdy \\ -\int_{\Omega}\epsilon \frac{\partial^2 u(x,y)}{\partial x^2 } v(x,y) dxdy -\int_{\Omega} \epsilon\frac{\partial^2 u(x,y)}{\partial y^2 } v(x,y) dxdy \\ l(v) = \int_{\Omega} f(x,y) v(x,y) dxdy \)
For the diffusion term, we apply the integration by parts formula, similar to the heat transport equations. We do it to reduce the degree of derivatives operating in the field of pollution concentration
\( a(u,v) = \int_{\Omega} \beta_x(x,y) \frac{\partial u(x,y)}{\partial x } dxdy + \int_{\Omega} \beta_y(x,y) \frac{\partial u(x,y)}{\partial y } dxdy \\ -\int_{\Omega} \epsilon \frac{\partial^2 u(x,y)}{\partial x^2 } v(x,y) dxdy -\int_{\Omega} \epsilon \frac{\partial^2 u(x,y)}{\partial y^2 } v(x,y) dxdy \\ = \int_{\Omega} \beta_x(x,y) \frac{\partial u(x,y) }{\partial x } dxdy +\int_{\Omega} \beta_y(x,y) \frac{\partial u(x,y)}{\partial y } dxdy \\ +\int_{\Omega} \epsilon \frac{\partial u(x,y) }{\partial x} \frac{\partial v(x,y)}{\partial x } dxdy +\int_{\Omega} \epsilon \frac{\partial u(x,y)}{\partial y } \frac{\partial v(x,y)}{\partial y } dxdy -\int_{\partial \Omega } \frac{\partial u }{\partial n } v dS \)
Usually, our advection-diffusion problem, similar to the heat transport problem, should be equipped with boundary conditions. We divide the boundary
\( \partial \Omega = \Gamma_D \cup \Gamma_N \) and we introduce
Dirichlet boundary condition, on
\( \Gamma_D \) part of the boundary \( \partial \Omega \), which tells us about the concentration of pollutants on the edge of the area \( u(x,y)=p(x,y) \textrm{ dla }(x,y) \in \Gamma_D \)
where \( p(x,y) \) is a given derivative of the concentration of pollutants normal to the boundary of the area.
\( \Gamma_N \) fragment on the shore
\( \frac{\partial u(x,y)}{\partial n}=g(x,y) \textrm{ dla }(x,y) \in \Gamma_N \)
where \( g(x,y) \) is the stream of pollutant concentration on the shore.
Diffusion convection equations for a large value difference between the advection vector and the diffusion coefficient (e.g. two orders of magnitude or more, i.e. \( \beta=(1,0), \epsilon=10^{-2} \)) \ require special stabilization methods as described in chapter five. For small differences in values, the traditional or isogeometric finite element method works fine.