Iterative solvers and optimal complexity of adaptive finite element methods

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…

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A parameter-free HDG method for linear elasticity with strongly symmetric stress

Lina Zhao: Thursday, 12th December at 11:00 In this talk, we present a parameter-free hybridizable discontinuous Galerkin (HDG) method of arbitrary polynomial orders for the linear elasticity problem, where the symmetry of stress is strongly imposed. The $H(\tdiv;\Omega)$-conforming space is used for the approximation of the displacement and the standard polynomial space is used for the approximation of the stress. The tangential trace of displacement acts as the Lagrange multiplier. The quasi-optimal approximation (up to data-oscillation term) is established for the $L^2$-error of stress and discrete $H^1$-error of displacement with $\lambda$-independent constants without requiring additional regularity assumption.  To guide adaptive mesh refinement, $\lambda$-robust a posteriori error estimator is derived. Several numerical experiments will be reported to demonstrate the performance of the proposed scheme.

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𝐻² conforming virtual element discretization of nondivergence form elliptic equations

Guillaume Bonnet: Thursday, 21st November at 11:00 The numerical discretisation of elliptic equations in nondivergence form is notoriously challenging, due to the lack of a notion of weak solutions based on variational principles. In many cases, there still is a well-posed variational formulation for such equations, which has the particularity of being posed in 𝐻², and therefore leads to a strong solution. Galerkin discretizations based on this formulation have been studied in the literature. Since 𝐻² conforming finite elements tend to be considered impractical, most of these discretizations are of discontinuous Galerkin type. On the other hand, it has been observed in the virtual element literature that the virtual element method provides a practical way to build 𝐻² conforming discretizations of variational problems. In this talk, I will describe a virtual element discretization of equations in nondivergence form. I will start with a simple linear model problem, and show how the 𝐻² conformity of the method allows for a particularly simple well-posedness and error analysis. I will then discuss the extension to equations with lower-order terms and with Hamilton-Jacobi-Bellman type nonlinearities, and present some numerical results.

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