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Register a new userAn adaptive finite element method for the inequality-constrained Reynolds equation
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We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem and approximated by a residual-based stabilized method. Based on our recent results on the classical obstacle problem, we present optimal a priori estimates and derive novel a posteriori error estimators. The method is implemented as a Nitsche-type finite element technique and shown in numerical computations to be superior to the usually applied penalty methods.
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Consider the electromagnetic scattering of a time-harmonic plane wave by an open cavity which is embedded in a perfectly electrically conducting infinite ground plane. This paper is concerned with the numerical solutions of the transverse electric and magnetic polarizations of the open cavity scattering problems. In each polarization, the scattering problem is reduced equivalently into a boundary value problem of the two-dimensional Helmholtz equation in a bounded domain by using the transparent boundary condition (TBC). An a posteriori estimate based adaptive finite element method with the perfectly matched layer (PML) technique is developed to solve the reduced problem. The estimate takes account both of the finite element approximation error and the PML truncation error, where the latter is shown to decay exponentially with respect to the PML medium parameter and the thickness of the PML layer. Numerical experiments are presented and compared with the adaptive finite element TBC method for both polarizations to illustrate the competitive behavior of the proposed method.
We design an adaptive unfitted finite element method on the Cartesian mesh with hanging nodes. We derive an hp-reliable and efficient residual type a posteriori error estimate on K-meshes. A key ingredient is a novel hp-domain inverse estimate which allows us to prove the stability of the finite element method under practical interface resolving mesh conditions and also prove the lower bound of the hp a posteriori error estimate. Numerical examples are included.
We introduce a new method for the numerical approximation of time-harmonic acoustic scattering problems stemming from material inhomogeneities. The method works for any frequency $omega$, but is especially efficient for high-frequency problems. It is based on a time-domain approach and consists of three steps: emph{i)} computation of a suitable incoming plane wavelet with compact support in the propagation direction; emph{ii)} solving a scattering problem in the time domain for the incoming plane wavelet; emph{iii)} reconstruction of the time-harmonic solution from the time-domain solution via a Fourier transform in time. An essential ingredient of the new method is a front-tracking mesh adaptation algorithm for solving the problem in emph{ii)}. By exploiting the limited support of the wave front, this allows us to make the number of the required degrees of freedom to reach a given accuracy significantly less dependent on the frequency $omega$, as shown in the numerical experiments. We also present a new algorithm for computing the Fourier transform in emph{iii)} that exploits the reduced number of degrees of freedom corresponding to the adapted meshes.
We propose a weak Galerkin(WG) finite element method for solving the one-dimensional Burgers equation. Based on a new weak variational form, both semi-discrete and fully-discrete WG finite element schemes are established and analyzed. We prove the existence of the discrete solution and derive the optimal order error estimates in the discrete $H^1$-norm and $L^2$-norm, respectively. Numerical experiments are presented to illustrate our theoretical analysis.
A lack of regularity in the solution of the porous medium equation poses a serious challenge in its theoretical and numerical studies. A common strategy in theoretical studies is to utilize the pressure formulation of the equation where a new variable called the mathematical pressure is introduced. It is known that the new variable has much better regularity than the original one and Darcys law for the movement of the free boundary can be expressed naturally in this new variable. The pressure formulation has not been used in numerical studies. The goal of this work is to study its use in the adaptive finite element solution of the porous medium equation. The MMPDE moving mesh strategy is employed for adaptive mesh movement while linear finite elements are used for spatial discretization. The free boundary is traced explicitly by integrating Darcys law with the Euler scheme. Numerical results are presented for three two-dimensional examples. The method is shown to be second-order in space and first-order in time in the pressure variable. Moreover, the convergence order of the error in the location of the free boundary is almost second-order in the maximum norm. However, numerical results also show that the convergence order of the method in the original variable stays between first-order and second-order in the $L^1$ norm or between 0.5th-order and first-order in the $L^2$ norm. Nevertheless, the current method can offer some advantages over numerical methods based on the original formulation for situations with large exponents or when a more accurate location of the free boundary is desired.
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