Main approaches to the content is built up around the design problem. It focuses on keywords such as consistency, convergence, stability, existence, uniqueness, and efficiency.The course covers the Fourier method, Energy method, normed vector spaces, bi-linear forms, Lp - and Sobolev spaces, weak derivatives, elliptic boundary value problems, hyperbolic and parabolic time-dependent initial value problems, linearisation for nonlinear problems, interpolants and finite elements, higher order methods and elements, stabilisation, error bounds.