In this article, we aim to recover locally conservative and $H(div)$ conforming fluxes for the linear Cut Finite Element Solution with Nitsches method for Poisson problems with Dirichlet boundary condition. The computation of the conservative flux in
the Raviart-Thomas space is completely local and does not require to solve any mixed problem. The $L^2$-norm of the difference between the numerical flux and the recovered flux can then be used as a posteriori error estimator in the adaptive mesh refinement procedure. Theoretically we are able to prove the global reliability and local efficiency. The theoretical results are verified in the numerical results. Moreover, in the numerical results we also observe optimal convergence rate for the flux error.
This paper is concerned with the problem of enhancing convection-cooling via active control of the incompressible velocity field, described by a stationary diffusion-convection model. This essentially leads to a bilinear optimal control problem. A ri
gorous proof of the existence of an optimal control is presented and the first order optimality conditions are derived for solving the control using a variational inequality. Moreover, the second order sufficient conditions are established to characterize the local minimizer. Finally, numerical experiments are conducted utilizing finite elements methods together with nonlinear iterative schemes, to demonstrate and validate the effectiveness of our control design.
In this paper we discuss a level set approach for the identification of an unknown boundary in a computational domain. The problem takes the form of a Bernoulli problem where only the Dirichlet datum is known on the boundary that is to be identified,
but additional information on the Neumann condition is available on the known part of the boundary. The approach uses a classical constrained optimization problem, where a cost functional is minimized with respect to the unknown boundary, the position of which is defined implicitly by a level set function. To solve the optimization problem a steepest descent algorithm using shape derivatives is applied. In each iteration the cut finite element method is used to obtain high accuracy approximations of the pde-model constraint for a given level set configuration without re-meshing. We consider three different shape derivatives. First the classical one, derived using the continuous optimization problem (optimize then discretize). Then the functional is first discretized using the CutFEM method and the shape derivative is evaluated on the finite element functional (discretize then optimize). Finally we consider a third approach, also using a discretized functional. In this case we do not perturb the domain, but consider a so-called boundary value correction method, where a small correction to the boundary position may be included in the weak boundary condition. Using this correction the shape derivative may be obtained by perturbing a distance parameter in the discrete variational formulation. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem.
We derive a residual based a-posteriori error estimate for the outer normal derivative of approximations to Poissons problem. By analyzing the solution of the adjoint problem, we show that error indicators in the bulk may be defined to be of higher o
rder than those close to the boundary, which lead to more economic meshes. The theory is illustrated with some numerical examples.
The well-known Prager-Synge identity is valid in $H^1(Omega)$ and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new inequality, that may be regarded as a genera
lization of the Prager-Synge identity, to be valid for piecewise $H^1(Omega)$ functions for diffusion problems. The inequality is proved to be identity in two dimensions. For nonconforming finite element approximation of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrated flux in $H(div; Omega)$ through a local element-wise scheme and that recovers a gradient in $H(curl;Omega)$ through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution.
In this work we study a residual based a posteriori error estimation for the CutFEM method applied to an elliptic model problem. We consider the problem with non-polygonal boundary and the analysis takes into account the geometry and data approximati
on on the boundary. The reliability and efficiency are theoretically proved. Moreover, constants are robust with respect to how the domain boundary cuts the mesh.
We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error esti
mates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.
