Partial Differential Equations in Two Space Dimensions

You can solve problems in two space dimensions using the same methods as applied in one dimension, except that the details become more complicated. The problems are strffer, and implicit methods are usually required. The problem of unwanted oscillations is more severe because added diffusion may smooth the solution in the flow direction, as desired, but it may also diffuse it in the direction perpendicular to flow, which is not desired. Careful treatment, though, allows for streamline diffusion that is, the extra diffusion is only added in the flow direction. These details have been handled by FEMLAB, which frees you to concentrate on the physics and the problem you want to solve. 

The trial functions are usually defined on an element. A square element in two dimensions with bilinear trial functions has four trial functions per element. The element covers 0 M I, 0 1, and the trial functions are  

The domain is covered with elements to represent a function on the entire domain. Consider the function 

---

MATLAB has a powerful toolbox for solution of linear and nonlinear partial differential equations which is called Partial Differential Equation (or PDE) TOOLBOX. This toolbox uses the finite element method for solution of partial differential equations in two space dimensions. The basic equation of this toolbox is the equation... 

Solution of parabolic partial differential equation in two space dimensions by the explicit method. 

---