A generalized finite element method for linear thermoelasticity

Abstract

Abstract. We propose and analyze a generalized finite element method designed for linear quasistatic thermoelastic systems with spatial multiscale coefficients. The method is based on the local orthogonal decomposition technique introduced by Målqvist and Peterseim (Math. Comp., 83(290): 2583Comp., 83(290): -2603Comp., 83(290): , 2014. We prove convergence of optimal order, independent of the derivatives of the coefficients, in the spatial H 1 -norm. The theoretical results are confirmed by numerical examples.

Computational Multiscale Methods for Linear Heterogeneous Poroelasticity

“…The goal is to construct a new function space with the same dimension as V H × Q H but with better approximation properties. For this, we follow the methodology of LOD [21,23] and, in particular, translate the results from thermoelasticity presented in [19] to the present setting.…”

“…With all tools in hand, we are now able to present the method proposed in [19] for thermoelasticity. Since the considered system involves time derivatives, the corrector problem needs to be solved in each time step.…”

“…The main goal of this section is to approximate the solution on a mesh of some feasible coarse scale of resolution independent of microscopic oscillations. Our construction is based on the concept of localized orthogonal decomposition (LOD) [21,3,23,14] and adapts ideas from thermoelasticity [19] and the heat equation [20].…”

“…Proof. As in the proof of the convergence of the multiscale method in [19], we split the errors in the displacement and pressure into two parts each, namely ρ n u := u n h − R 1 ms u n h , η n u := R 1 ms u n h − u n ms , ρ n p := p n h − R 2 ms p n h , η n p := R 2 ms p n h − p n ms .…”

Computational multiscale methods for linear poroelasticity with high contrast

“…[13,16,17,23,25]. This method was applied in [22] to linear heterogeneous thermoelasticity and optimal first-order convergence of the fully discretized system based on LOD and an implicit Euler discretization in time was proven. This approach was transferred to the present poroelastic setting in [1].…”