Ani Miraci: Tuesday, 17th December at 11:00
Finite element methods (FEMs) are often used to discretize second-order elliptic partial differential equations (PDEs). While standard FEMs rely on underlying uniform meshes, adaptive FEMs (AFEMs) drive the local mesh-refinement to capture potential singularities of the (unknown) PDE solution (stemming, e.g., from the data or the domain geometry). Crucially, adaptivity is steered by reliable a posteriori error control, often encoded in the paradigm SOLVE — ESTIMATE — MARK — REFINE.
AFEMs allow to obtain optimal rates of convergence with respect to the number of degrees of freedom (an improvement to standard FEMs). However, in terms of computational costs, an adaptive algorithm is inherently cumulative in nature: an initial coarse mesh is used as input and exact finite element solutions need to be computed on consecutively refined meshes before a desired accuracy can be ensured. Thus, in practice, one strives instead to achieve optimal complexity, i.e., optimal rate of convergence with respect to the overall computational cost.
The core ingredient needed for optimal complexity consists in the use of appropriate iterative solvers to be integrated as the SOLVE module within the adaptive algorithm. More precisely, one requires:
(i) a solver whose each iteration is: (a) of linear complexity and (b) contractive;
(ii) a-posteriori-steered solver-stopping criterion which allows to discern and balance discretization and solver error;
(iii) nested iteration, i.e., the last computed solver-iterate is used as initial guess in the newly-refined mesh.
First, we develop an optimal local multigrid for the context of symmetric linear elliptic second order PDEs and a finite element discretization with a fixed polynomial degree p and a hierarchy of bisection-generated meshes with local size h. The solver contracts the algebraic error hp-robustly and comes with a built-in a posteriori estimator equivalent to the algebraic error.
Second, the overall adaptive algorithm is then shown to be convergent for any choice of adaptivity parameters (parameter-robust convergence), and to be of optimal complexity for small parameters.
Third, we provide extensions to certain non-symmetric or non-linear PDEs, where a nested iterative solver structure fits into the general analytical framework of the linear symmetric case.
Numerical experiments highlight the theoretical results of optimal complexity for adaptivity with iterative solvers and emphasize the practical relevance and gains of such numerical simulations.
