Advanced finite element methods and applications by Thomas Apel, Olaf Steinbach

By Thomas Apel, Olaf Steinbach

This quantity on a few fresh facets of finite point tools and their purposes is devoted to Ulrich Langer and Arnd Meyer at the social gathering in their sixtieth birthdays in 2012. Their paintings combines the numerical research of finite aspect algorithms, their effective implementation on cutting-edge architectures, and the collaboration with engineers and practitioners. during this spirit, this quantity includes contributions of former scholars and collaborators indicating the extensive variety in their pursuits within the thought and alertness of finite aspect methods.

Topics hide the research of area decomposition and multilevel equipment, together with hp finite parts, hybrid discontinuous Galerkin equipment, and the coupling of finite and boundary aspect equipment; the effective answer of eigenvalue difficulties concerning partial differential  equations with functions in electric engineering and optics; and the answer of direct and inverse box difficulties in reliable mechanics.

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DD Preconditioning for High Order Hybrid DG Methods 29 2 HDG Discretization Let Ω ⊂ R3 be a polyhedral domain. Let T = {T } be a conforming triangulation of Ω consisting of shape regular tetrahedral elements. With F = {F} we denote the set of all faces, and FT are the faces of the element T . As usual hT = diam T is the local mesh-size. We consider the Dirichlet problem of the Poisson equation problem, namely −Δ u = f in Ω , u = 0 on ∂ Ω , with the source f ∈ L2 (Ω ). We define the pth order hybrid discontinuous Galerkin finite element space VN := P p (T ) × P p(F ) := ∏ P p(T ) × ∏ P p(F), T ∈T F∈F its subspace VN,0 = {(u, λ ) ∈ VN : λ = 0 on ∂ Ω }, and the hybrid discontinuous Galerkin (HDG) method as: find (uN , λN ) ∈ VN,0 : A(uN , λN ; v, μ ) = ( f , v)L2 (Ω ) ∀ (v, μ ) ∈ VN,0 .

Usually α is chosen on the safe side. We will see in the numerical examples that the condition number does increase with α . In this paper we prove 2 DD Preconditioning for High Order Hybrid DG Methods 33 quasi-optimal condition numbers for the presented stabilization, it does not carry over to the weighted L2 -stabilization. The benefit is two-fold, on one side the method is guaranteed to be stable, for any α > |FT |, on the other side the condition number is proven to have only poly-logarithmic growth.

This paper is concerned with the construction and analysis of domain decomposition methods for the Hybrid Discontinuous Galerkin (HDG) method. We consider one element as sub-domain, and the coarse grid problem consists of mean values on element interfaces. de T. Apel & O. , LNACM 66, pp. 27–56. com 28 J. Sch¨oberl and C. Lehrenfeld robustness with respect to the mesh-size, and a poly-logarithmic growth of the condition number with the polynomial order p. There is now an established literature on high order finite element methods, from the more theoretical point of view as well as from an applied one [13, 26, 41, 43].

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