Adaptive Finite Elements in the Discretization of Parabolic Problems
Adaptivity is a crucial tool in state-of-the-art scientific computing. However, its theoretical foundations are only understood partially and are subject of current research. This self-contained work provides theoretical basics on partial differential equations and finite element discretizations before focusing on adaptive finite element methods for time dependent problems. In this context, aspects of temporal adaptivity and error control are considered in particular. Based on the gained insights, a specific adaptive algorithm is designed and analyzed thoroughly. Most importantly, it is proven that the presented adaptive method terminates within any demanded error tolerance. Moreover, the developed algorithm is analyzed from a numerical point of view and its performance is compared to well-known standard methods. Finally, it is applied to the real-life problem of concrete carbonation, where two different discretizations are compared.
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adaptive algorithm approximation ASTFEM Banach space bilinear form carbonation depth Cauchy-Schwarz inequality CLASSIC coarsen time error coarsen-time indicator coarsened mesh coarsening error coarsening strategy compute concrete carbonation conforming triangulation consider convergence coupled cretization current time step decoupled discretization deﬁne deﬁnition denote derive discretization error element markers element space Vn elliptic problem employ energy norm equation error control error estimate error indicator Figure ﬁnd ﬁnite finite element space ﬁrst G Vn Galerkin Ginit given global grid Hence Hilbert space implies interpolation iterations L2-norm L2-projection Lagrange interpolation Lemma linear MARKCOARSEN marked elements mesh G minimal time step Moreover n-th time step nodes parameter particularly refining the mesh residual respectively right hand side Robin boundary condition satisﬁed Section sequence SOLVE spatial STADAPTATION step size step sizes symmetric bilinear form terminates Theorem tion TOL0 tolerance TOL TOLf Un_1 weak derivative weak formulation while-loop