Partial Differential Equations(PDEs) play an important role in science and engineering applications, such as the propagation of heat or sound, fluid flow, finite elasticity, electrodynamics, cancer modeling, etc. In general, the solution of PDEs can be very challenging, depending on the type of equation, the number of independent variables, the boundary and initial conditions, and nonlinearities. Even if we know the theoretical existence and uniqueness of the solution, obtaining analytical solutions may not be possible. The only way to get approximate solutions is by using numerical techniques. The finite element (FE) and discontinuous Galerkin (dG) method provide a formalism for generating discrete algorithms for approximating the solutions of differential equations. The finite element method (FEM) has a very strong foundation in the theory of functional analysis, and this gives us access to powerful tools when analyzing the error of the approximate solution. The variational approach also gives a solid mathematical foundation and make the error analysis more systematic. Generally speaking, the finite element method is the method of choice in all types of analysis for elliptic equations in complex domains. 

This workshop offers an introduction to some important numerical methods for linear and non-linear partial differential equations, and various numerical discretization techniques for their solution, namely finite element methods (FEM). It starts with a brief introduction to Sobolev spaces and elliptic scalar problems and continues with an explanation of finite element spaces and estimates for the interpolation error. This workshop aims to show how this abstraction can be used not only to analyze the stability and convergence of FE methods, but also to understand the realization of the abstraction into concrete computer code using the numerical softwares.