67 research outputs found
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Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier-Stokes equations
Recently, it was understood how to repair a certain L2-orthogonality of
discretely-divergence-free vector fields and gradient fields such that the
velocity error of inf-sup stable discretizations for the incompressible
Stokes equations becomes pressure-independent. These new pressure-robust
Stokes discretizations deliver a small velocity error, whenever the
continuous velocity field can be well approximated on a given grid. On the
contrary, classical inf-sup stable Stokes discretizations can guarantee a
small velocity error only, when both the velocity and the pressure field can
be approximated well, simultaneously. In this contribution,
pressure-robustness is extended to the time-dependent Navier-Stokes
equations. In particular, steady and time-dependent potential flows are shown
to build an entire class of benchmarks, where pressure-robust discretizations
can outperform classical approaches significantly. Speedups will be explained
by a new theoretical concept, the discrete Helmholtz projector of an inf-sup
stable discretization. Moreover, different discrete nonlinear convection
terms are discussed, and skew-symmetric pressure-robust discretizations are
proposed
Pressure-robustness and discrete Helmholtz projectors in mixed finite element methods for the incompressible Navier--Stokes equations
Recently, it was understood how to repair a certain L2-orthogonality of discretely-divergence-free vector fields and gradient fields such that the velocity error of inf-sup stable discretizations for the incompressible Stokes equations becomes pressure-independent. These new 'pressure-robust' Stokes discretizations deliver a small velocity error, whenever the continuous velocity field can be well approximated on a given grid. On the contrary, classical inf-sup stable Stokes discretizations can guarantee a small velocity error only, when both the velocity and the pressure field can be approximated well, simultaneously. In this contribution, 'pressure-robustness' is extended to the time-dependent Navier--Stokes equations. In particular, steady and time-dependent potential flows are shown to build an entire class of benchmarks, where pressure-robust discretizations can outperform classical approaches significantly. Speedups will be explained by a new theoretical concept, the 'discrete Helmholtz projector' of an inf-sup stable discretization. Moreover, different discrete nonlinear convection terms are discussed, and skew-symmetric pressure-robust discretizations are proposed
On spurious oscillations due to irrotational forces in the Navier--Stokes momentum balance
This contribution studies the influence of the pressure on the velocity error in finite element discretisations of the Navier--Stokes equations. Three simple benchmark problems that are all close to real-world applications convey that the pressure can be comparably large and is not to be underestimated. For widely used finite element methods like the Taylor--Hood finite element method, such relatively large pressures can lead to spurious oscillations and arbitrarily large errors in the velocity, even if the exact velocity is in the ansatz space. Only mixed finite element methods, whose velocity error is pressure-independent, like the Scott--Vogelius finite element method can avoid this influence
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Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM
Equilibration error estimators have been shown to commonly lead to very
accurate guaranteed error bounds in the a posteriori error control of finite
element methods for second order elliptic equations. Here, we extend previous
results by the design of equilibrated fluxes for higher-order finite element
methods with nonconstant coefficients and illustrate the favourable
performance of different variants of the error estimator within two
deterministic benchmark settings. After the introduction of the respective
parametric problem with stochastic coefficients and the stochastic Galerkin
FEM discretisation, a novel a posteriori error estimator for the stochastic
error in the energy norm is devised. The error estimation is based on the
stochastic residual and its decomposition into approximation residuals and a
truncation error of the stochastic discretisation. Importantly, by using the
derived deterministic equilibration techniques for the approximation
residuals, the computable error bound is guaranteed for the considered class
of problems. An adaptive algorithm allows the simultaneous refinement of the
deterministic mesh and the stochastic discretisation in anisotropic Legendre
polynomial chaos. Several stochastic benchmark problems illustrate the
efficiency of the adaptive process
Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem
Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the testfunctions, some explicit interpolation of the virtual testfunctions is needed that can be evaluated pointwise everywhere. The standard discretisation via an L2 -bestapproximation does not preserve the divergence and so destroys the orthogonality between divergence-free testfunctions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness another divergence-preserving reconstruction is suggested based on Raviart--Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes
Pressure-robustness in the context of optimal control
This paper studies the benefits of pressure-robust discretizations in the scope of optimal control of incompressible flows. Gradient forces that may appear in the data can have a negative impact on the accuracy of state and control and can only be correctly balanced if their L2-orthogonality onto discretely divergence-free test functions is restored. Perfectly orthogonal divergence-free discretizations or divergence-free reconstructions of these test functions do the trick and lead to much better analytic a priori estimates that are also validated in numerical examples
Robust equilibration a posteriori error estimation for convection-diffusion-reaction problems
We study a posteriori error estimates for convection-diffusion-reaction problems with possibly dominating convection or reaction and inhomogeneous boundary conditions. For the conforming FEM discretisation with streamline diffusion stabilisation (SDM), we derive robust and efficient error estimators based on the reconstruction of equilibrated fluxes in an admissible discrete subspace of H (div, Ω). Error estimators of this type have become popular recently since they provide guaranteed error bounds without further unknown constants. The estimators can be improved significantly by some postprocessing and divergence correction technique. For an extension of the energy norm by a dual norm of some part of the differential operator, complete independence from the coefficients of the problem is achieved. Numerical benchmarks illustrate the very good performance of the error estimators in the convection dominated and the singularly perturbed cases
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Guaranteed energy error estimators for a modified robust Crouzeix-Raviart Stokes element
This paper provides guaranteed upper energy error bounds for a modified
lowest-order nonconforming Crouzeix-Raviart finite element method for the
Stokes equations. The modification from [A. Linke 2014, On the role of the
Helmholtz-decomposition in mixed methods for incompressible flows and a new
variational crime] is based on the observation that only the divergence-free
part of the right-hand side should balance the vector Laplacian. The new
method has optimal energy error estimates and can lead to errors that are
smaller by several magnitudes, since the estimates are pressure-independent.
An efficient a posteriori velocity error estimator for the modified method
also should involve only the divergence-free part of the right-hand side.
Some designs to approximate the Helmholtz projector are compared and verified
by numerical benchmark examples. They show that guaranteed error control for
the modified method is possible and almost as sharp as for the unmodified
method
Local equilibration error estimators for guaranteed error control in adaptive stochastic higher-order Galerkin FEM
Equilibration error estimators have been shown to commonly lead to very accurate guaranteed error bounds in the a posteriori error control of finite element methods for second order elliptic equations. Here, we extend previous results by the design of equilibrated fluxes for higher-order finite element methods with nonconstant coefficients and illustrate the favourable performance of different variants of the error estimator within two deterministic benchmark settings. After the introduction of the respective parametric problem with stochastic coefficients and the stochastic Galerkin FEM discretisation, a novel a posteriori error estimator for the stochastic error in the energy norm is devised. The error estimation is based on the stochastic residual and its decomposition into approximation residuals and a truncation error of the stochastic discretisation. Importantly, by using the derived deterministic equilibration techniques for the approximation residuals, the computable error bound is guaranteed for the considered class of problems. An adaptive algorithm allows the simultaneous refinement of the deterministic mesh and the stochastic discretisation in anisotropic Legendre polynomial chaos. Several stochastic benchmark problems illustrate the efficiency of the adaptive process
Guaranteed energy error estimators for a modified robust Crouzeix--Raviart Stokes element
This paper provides guaranteed upper energy error bounds for a modified lowest-order nonconforming Crouzeix--Raviart finite element method for the Stokes equations. The modification from [A. Linke 2014, On the role of the Helmholtz-decomposition in mixed methods for incompressible flows and a new variational crime] is based on the observation that only the divergence-free part of the right-hand side should balance the vector Laplacian. The new method has optimal energy error estimates and can lead to errors that are smaller by several magnitudes, since the estimates are pressure-independent. An efficient a posteriori velocity error estimator for the modified method also should involve only the divergence-free part of the right-hand side. Some designs to approximate the Helmholtz projector are compared and verified by numerical benchmark examples. They show that guaranteed error control for the modified method is possible and almost as sharp as for the unmodified method
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