Mixed and hybrid finite element methods
Mixed and hybrid finite element methods
On the implementation of mixed methods as nonconforming methods for second-order elliptic problems
Mathematics of Computation
Convergence of Adaptive Finite Element Methods
SIAM Review
Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
A Local A Posteriori Error Estimator Based on Equilibrated Fluxes
SIAM Journal on Numerical Analysis
Space-Time adaptive algorithm for the mixed parabolic problem
Numerische Mathematik
Convergence of Adaptive Discontinuous Galerkin Approximations of Second-Order Elliptic Problems
SIAM Journal on Numerical Analysis
A posteriori error estimators for locally conservative methods of nonlinear elliptic problems
Applied Numerical Mathematics
SIAM Journal on Numerical Analysis
A Posteriori Error Estimation for Discontinuous Galerkin Finite Element Approximation
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
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We derive a posteriori error estimates for the discretization of the heat equation in a unified and fully discrete setting comprising the discontinuous Galerkin, various finite volume, and mixed finite element methods in space and the backward Euler scheme in time. Extensions to conforming and nonconforming finite element spatial discretizations are also outlined. Our estimates are based on a $H^1$-conforming reconstruction of the potential, continuous and piecewise affine in time, and a locally conservative $\mathbf{H}(\mathrm{div})$-conforming reconstruction of the flux, piecewise constant in time. They yield a guaranteed and fully computable upper bound on the error measured in the energy norm augmented by a dual norm of the time derivative. Local-in-time lower bounds are also derived; for nonconforming methods on time-varying meshes, the lower bounds require a mild parabolic-type constraint on the meshsize.