5. Adaptive space-time isogeometric analysis for parabolic evolution problems

Abstract

The paper is concerned with locally stabilized space-time IgA approximations to initial boundary value problems of the parabolic type. Originally, similar schemes (but weighted with a global mesh parameter) were presented and studied by U. Langer, M. Neumüller, and S. Moore (2016). The current work devises a localised version of this scheme. The localization of the stabilizations enables local mesh refinement that is one of the main ingredients of adaptive algorithms. We establish coercivity, boundedness, and consistency of the corresponding bilinear form. Using these fundamental properties together with the corresponding approximation error estimates for B-splines, we show that the space-time IgA solutions generated by the new scheme satisfy asymptotically optimal a priori discretization error estimates. The adaptive mesh refinement algorithm proposed in the paper is based on a posteriori error estimates of the functional type that has been rigorously studied in earlier works by S. Repin (2002) and U. Langer, S. Matculevich, and S. Repin (2017). Numerical results presented in the paper confirm the improved convergence of global approximation errors. Moreover, these results also confirm local efficiency of the error indicators produced by the error majorants.Keywords. parabolic initial-boundary value problems, locally stabilized space-time isogeometric analysis, a priori and a posteriori estimates of approximation errors.

Space‐Time Isogeometric Analysis of Parabolic Diffusion Problems in Moving Spatial Domains

“…We assume that p ≥ 2, and we use splines of the highest smoothness that belong to C p−1 . For fixed spatial domains, scheme (2) was introduced in [2]. Following this work, we can show that the bilinear form…”

“…In the numerical example, we measure the error using different norms. In addition to |||v h ||| h , we consider ∇ x e Q and |||e||| L := ( div x (ν(x, t)∇ x e) 2 Q + ∂ t e 2 Q + ν 1/2 ∇ x e 2 Σ T ) 0.5 , which are controlled by the majorants M I (u h , y h ) and M II (u h , y h , w h ) (see [2,3]), and by the error identity (see [4])…”

“…For fixed spatial domains, scheme (2) was introduced in [2]. We assume that p ≥ 2, and we use splines of the highest smoothness that belong to C p−1 .…”

“…In addition to |||v h ||| h , we consider ∇ x e Q and |||e||| L := ( div x (ν(x, t)∇ x e) 2 Q + ∂ t e 2 Q + ν 1/2 ∇ x e 2 Σ T ) 0.5 , which are controlled by the majorants M I (u h , y h ) and M II (u h , y h , w h ) (see [2,3]), and by the error identity (see [4]) In addition to |||v h ||| h , we consider ∇ x e Q and |||e||| L := ( div x (ν(x, t)∇ x e) 2 Q + ∂ t e 2 Q + ν 1/2 ∇ x e 2 Σ T ) 0.5 , which are controlled by the majorants M I (u h , y h ) and M II (u h , y h , w h ) (see [2,3]), and by the error identity (see [4])…”

Space-time hp finite elements for heat evolution in laser-based additive manufacturing

“…CG approaches tend to be more efficient for a given number of unknowns; therefore, they are suitable for local mesh refinement in space-time discretizations. Different formulations have been analyzed, such as [13,14], and in particular [15], where they use truncated hierarchical B-Splines to develop an adaptive space-time formulation with higher continuity. Spacetime finite element methods are also used to compute problems with moving boundaries [16,17].…”