We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on
-energy estimates on discrete functions in physical space....

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on
-energy estimates on discrete functions in physical space....

We analyze Euler-Galerkin approximations (conforming finite elements in space and implicit Euler in time) to coupled PDE systems in which one dependent variable, say $u$, is governed by an elliptic equation and the other, say $p$, by a parabolic-like equation. The underlying application is the poroelasticity system within the quasi-static assumption. Different polynomial orders are used for the $u$- and $p$-components to obtain optimally convergent a priori bounds for all the terms in the error energy norm....

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is...

We analyze a two-stage implicit-explicit Runge–Kutta scheme for time discretization of advection-diffusion equations. Space discretization uses continuous, piecewise affine finite elements with interelement gradient jump penalty; discontinuous Galerkin methods can be considered as well. The advective and stabilization operators are treated explicitly, whereas the diffusion operator is treated implicitly. Our analysis hinges on
-energy estimates on discrete functions in physical space....

A continuous finite element method to approximate Friedrichs' systems is
proposed and analyzed. Stability is achieved by penalizing the jumps
across mesh
interfaces of the normal derivative of some components of the discrete solution.
The convergence analysis leads to optimal convergence rates
in the graph norm and suboptimal of order ½ convergence rates in
the
-norm. A variant of the method specialized to
Friedrichs' systems associated with elliptic PDE's in mixed form and
reducing...

We analyze Euler-Galerkin approximations (conforming finite elements in
space and implicit Euler in time) to
coupled PDE systems in which one dependent
variable, say , is governed by an elliptic equation and the other,
say , by a parabolic-like equation. The underlying application is the
poroelasticity system within the quasi-static assumption. Different
polynomial orders are used for the - and -components to
obtain optimally convergent bounds for all
the terms in the error energy norm.
Then,...

The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off...

We analyze residual and hierarchical a posteriori error estimates for nonconforming finite element approximations of elliptic problems with variable coefficients. We consider a finite volume box scheme equivalent to a nonconforming mixed finite element method in a Petrov–Galerkin setting. We prove that all the estimators yield global upper and local lower bounds for the discretization error. Finally, we present results illustrating the efficiency of the estimators, for instance, in the simulation...

This paper derives upper and lower bounds for the ${\ell}^{p}$-condition
number of the stiffness matrix resulting from the finite element
approximation of a linear, abstract model problem. Sharp estimates in
terms of the meshsize are obtained. The theoretical results are
applied to finite element approximations of elliptic PDE's in
variational and in mixed form, and to first-order PDE's approximated
using the Galerkin–Least Squares technique or by
means of a non-standard Galerkin technique in
...

We investigate unilateral contact problems with cohesive forces, leading to
the constrained minimization of a possibly nonconvex functional. We
analyze the mathematical structure of the minimization problem.
The problem is reformulated in terms of a three-field augmented
Lagrangian, and sufficient conditions for the existence of a local
saddle-point are derived. Then, we derive and analyze mixed finite
element approximations to the stationarity conditions of the three-field
augmented Lagrangian....

We analyze residual and hierarchical
error estimates for nonconforming finite element
approximations of elliptic problems with variable coefficients.
We consider a finite volume box scheme equivalent to
a nonconforming mixed finite element method in a Petrov–Galerkin
setting. We prove that
all the estimators yield global upper and local lower bounds for the discretization
error. Finally, we present results illustrating the efficiency of the
estimators, for instance, in the simulation of Darcy...

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