Traditional, low order, finite element discretizations are well suited to resolve complex topologies and curvilinear geometries. The corresponding rates of convergence are limited by the polynomial order, and regularity of the solution. The regularity is limited by singularities coming from non-convex geometries and material interfaces but also regions with high gradients such as boundary layers perceived by the computer in the preasymptotic range as singularities. In presence of problems with large geometrical or material contrasts, they "lock" (100 % error). For wave propagation problems, they suffer from large dispersion (phase) errors making solution of problems with large wave numbers impossible.
Spectral methods do not lock for singularly perturbed problems, and deliver exponential convergence, provided the solution is analytic up to boundary, i.e. no singularities are present on the boundary. They do not suffer from dispersion error for wave propagation. If the solution is, however, singular on the boundary or material interfaces, the advantage of using spectral methods is lost - the convergence slows down to algebraic rates again. They behave also very badly in the preasymptotic range if the meshes do not reflect the structure of the solution. For complex curvilinear geometries, meshes are difficult to generate.
hp Finite Element Methods combine advantages of low order and spectral methods. The possibility of varying both element size h and polynomial order p allows for capturing small geometrical scales and avoiding locking or dispersion errors. Properly designed hp-adaptive schemes deliver exponential convergence (error vs. cost) for both regular and irregular (singular) solutions. The main price to pay is the complexity of the methodology and corresponding codes.
The week-long short course presents fundamental components of hp Finite Element Methods. Four to six hours of lectures on theory are illustrated with numerical experiments and ``hands on'' experience with a 2D hp code. The course focuses on linear problems with an outlook at some non-linear examples from CFD and includes our most recent results on Discontinuous Petrov-Galerkin (DPG) methods with optimal test functions. A tentative program of the course is as follows.
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The majority of presented material is covered in [1,2]. We will help to install the 1D and 2D codes from [1] on the participants' laptops. The laptops have to operate under LINUX and must have the Intel Fortran 90 and C compilers installed.
[1] L. Demkowicz, L., ``Computing with hp Finite Elements. I.One- and Two-Dimensional Elliptic and Maxwell Problems'', Chapman & Hall/CRC Press, Taylor and Francis, October 2006.
[2] L. Demkowicz, J. Kurtz, D. Pardo, M. Paszynski, W. Rachowicz and A. Zdunek, ``Computing with hp Finite Elements. II. Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications'', Chapman & Hall/CRC Press, Taylor and Francis, August 2007.