p-robust multilevel algebraic error estimator & solver

Main results: multilevel a posteriori estimator of the algebraic error; guaranteed (reliable) and efficient; polynomial-degree robust; gives rise to a multilevel iterative algebraic solver with contraction independent of p; equivalence estimator – solver; Construction: global coarsest-level solve of lowest-order (p=1); local patchwise contributions from other levels; corresponds to V-cycle geometric…

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Adaptive inexact semismooth Newton method for variational inequalities

Main results: guaranteed upper bound on the total error; distinction of different error components (spatial discretization, temporal discretization, semismooth Newton linearization, algebraic solver); design of an adaptive inexact semismooth Newton method for variational inequalities; application to unsteady nonlinear coupled degenerate variational inequalities (compositional two-phase flow in porous media with exchange…

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Goal-oriented adaptivity with inexact solvers

Main results: quantity of interest given by a volumetric (source) or a surface (flux) term; same discretization of both primal and dual problems; inexact algebraic solvers for both primal and dual problems; guaranteed upper bound on the goal error unified framework covering basically any numerical method (conforming, nonconforming, and discontinuous…

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Polytopal meshes and complex porous media flows

Main results: general polytopal meshes; easy-to-implement and fast-to-run a posteriori error estimates given by a simple matrix–vector multiplication; works for any lowest-order locally conservative method; guaranteed upper bound on the total error; distinction of different error components (spatial discretization, temporal discretization, linearization, algebraic solver); unsteady nonlinear coupled degenerate problems (multiphase…

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Guaranteed bounds on the error reduction and hp adaptivity with inexact solvers

Main results: computable guaranteed bound on the error reduction factor (ratio of the errors between two successive hp refinements); homogeneous Neumann local problems via mixed finite elements to obtain error upper bound; homogeneous Dirichlet local problems via conforming finite elements for hp decision; hp-adaptive strategy with numerically exponential convergence rates;…

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Stable broken H^1 and H(div) polynomial extensions

Main results: H^1-stable polynomial extension on a tetrahedron; H(div)-stable polynomial extension on a tetrahedron; stable broken H^1 polynomial extension on a patch of thetrahedra; stable broken H(div) polynomial extension on a patch of thetrahedra; polynomial-degree-robust efficiency of H^1-nonconforming methods; polynomial-degree-robust efficiency of H(div)-nonconforming methods. Details: paper with Alexandre Ern (preprint)…

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Local space-time efficiency

Main results (heat equation): inf-sup equality with constant 1; any order conforming continuous Galerkin in space; any order discontinuous Galerkin in time; guranteed reliability (upper bound); efficiency local with respect to both space and time; robustness with respect to polynomial degrees both in space and time; arbitrary space mesh refinement…

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Guaranteed eigenvalue and eigenvector bounds

Main results: guaranteed upper and lower bounds for an arbitrary simple eigenvalue, multiple eigenvalue, or a cluster of eigenvalues; guaranteed energy error bounds on the associated eigenvector(s); efficiency (optimal convergence rate) and polynomial-degree robustness; multiplicative constant tending to one under an elliptic regularity assumption on the corresponding source problem; inexact…

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Localization of the W^{-1,q} and distance norms

  Main results: dual norm of any bounded linear functional on the Sobolev space W01,p localizes under an orthogonality condition: equals the lq sum of local contributions; estimates taking into account the violation of the orthogonality condition; distance to the Sobolev space H01 localizes: equals the l2 sum of local…

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