Variational formulation of boundary value problems The Finite Element Method (FEM): definition, basic properties hp- Finite Elements in one dimension hp- Finite Elements in two dimensions Finite… Expand

The hp-version of the discontinuous Galerkin finite element method for second-order partial differential equations with nonnegative characteristic form is considered, and an hp-optimal error bound is derived in the hyperbolic case and in the self-adjoint elliptic case.Expand

In Chap. 3 we transformed strongly elliptic boundary value problems of second order in domains \( \Omega \subset \mathbb{R}^3\) into boundary integral equations. These integral equations were… Expand

The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem.Expand

An interpolation technique in which the sample set is incremented as the polynomial dimension increases, leading therefore to a minimal amount of PDE solving, which is based on the standard principle of tensorisation of a one-dimensional interpolation scheme and sparsification.Expand

Partial differential equations with random input data, such as random loadings and coefficients, are reformulated as parametric, deterministic PDEs on parameter spaces of high, possibly infinite dimension to derive representation of the random solutions' laws on infinite-dimensional parameter spaces in terms of ‘generalized polynomial chaos’ (GPC) series.Expand

The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of theGalerkin method and it is shown that the hp's spectral convergence gives spectral convergence in problems with smooth time dependence.Expand

Parametric partial dierential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic… Expand