Polytopal meshes and complex porous media flows

Main results: general polytopal meshes; easy-to-implement and fast-to-run 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 multicomponent flows in…

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Guaranteed bounds on the error reduction and hp adaptivity

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: preprint with Alexandre Ern and…

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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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Laplace eigenvalues and eigenvectors bounds

Main results: guaranteed upper and lower bounds for an arbitrary simple eigenvalue; guaranteed energy error bounds on the associated eigenvector; efficiency (optimal convergence rate) and polynomial-degree robustness; multiplicative constant tending to one under an elliptic regularity assumption on the corresponding source problem; inexact algebraic solvers taken into account; unified framework…

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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 W^{1,p}_0 localizes under an orthogonality condition: it is equivalent to the l^q sum of local contributions; estimates taking into account the violation of the orthogonality condition; distance to the Sobolev space H^1_0 localizes: it is equivalent…

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Estimating and localizing the algebraic and total errors

Main results: total, algebraic, and discretization errors; guaranteed upper and lower bounds; arbitrary iterative algebraic solver; estimating the local distribution of the errors over the computational domain; safe stopping criteria for iterative algebraic solvers: the algebraic error will lie below the discretization one. Details: paper with Jan Papež and Zdeněk…

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Stopping criteria for domain decomposition methods

Main results: guaranteed error upper bound on each DD iteration; distinguishes the domain decomposition error, the spatial discretization error, and possibly the temporal discretization error; important savings in terms of the number of domain decomposition iterations; optimized Robin and Ventcell transmission conditions; global-in-time formulation that allows for use of local…

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Potential and flux reconstructions a posteriori error estimates

Main results: potential and equilibrated flux reconstruction estimates in a unified framework; inhomogeneous Dirichlet and Neumann boundary conditions; spatially varying polynomial degree; mixed rectangular-triangular grids; combination with the H(div)-liftings paper: robustness with respect to the number of hanging nodes; asymptotic exactness observed for smooth solutions; hp-adaptivity strategy numerically leading to…

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