Volume 18, Issue 2

In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯H⁢(D){\mathcal{T}_{H}(D)} with small scale of sparse collocation points η⁢(L,N){\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯h⁢(D){\mathcal{T}_{h}(D)} using large scale of sparse collocation points η⁢(ℓ,N){\eta(\ell,N)} (where η⁢(L,N),η⁢(ℓ,N){\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L,ℓ{L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h≈H3,η⁢(ℓ,N)≈(η⁢(L,N))3{{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}}, the novel two-level discretization method achieves the same convergence accuracy in norm ∥⋅∥ℒρ2⁢(Γ)⊗ℒ2⁢(D){\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} (ℒρ2⁢(Γ){\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ2{\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯h⁢(D){\mathcal{T}_{h}(D)} and large scale collocation points η⁢(ℓ,N){\eta(\ell,N)} in random spaces.

We consider a reduced Galerkin least-squares finite element method for the Oldroyd-B model of viscoelastic fluid flows. Model problems considered are the flow past a planar channel and a 4-to-1 contraction problems. An a priori error estimate for the reduced Galerkin least-squares method is derived and numerical results supporting the estimate are presented.

In this short note we analyse a connection between the exponentially graded and a generalisation of the class of S-type meshes for singularly perturbed problems.

The convergence of GMRES for solving linear systems can be influenced heavily by the structure of the right-hand side. Within the solution of eigenvalue problems via inverse iteration or subspace iteration, the right-hand side is generally related to an approximate invariant subspace of the linear system. We give detailed and new bounds on (block) GMRES that take the special behavior of the right-hand side into account and explain the initial sharp decrease of the GMRES residual. The bounds motivate the use of specific preconditioners for these eigenvalue problems, e.g., tuned and polynomial preconditioners, as we describe. The numerical results show that the new (block) GMRES bounds are much sharper than conventional bounds and that preconditioned subspace iteration with either a tuned or polynomial preconditioner should be used in practice.

An optimally convergent (with respect to the regularity) quadratic finite element method for the two-dimensional obstacle problem on simplicial meshes is studied in [14]. There was no analogue of a quadratic finite element method on tetrahedron meshes for the three-dimensional obstacle problem. In this article, a quadratic finite element enriched with element-wise bubble functions is proposed for the three-dimensional elliptic obstacle problem. A priori error estimates are derived to show the optimal convergence of the method with respect to the regularity. Further, a posteriori error estimates are derived to design an adaptive mesh refinement algorithm. A numerical experiment illustrating the theoretical result on a priori error estimates is presented.

We first derive some explicit bounds on the spectra of generalized non-symmetric singular or nonsingular saddle point matrices. Then we propose two new nonsingular preconditioners for solving generalized singular saddle point problems, and show that GMRES determines a solution without breakdown when applied to the resulting preconditioned systems with any initial guess. Furthermore, the detailed spectral properties of the preconditioned systems are analyzed. The nonsingular preconditioners are also applied to solve the singular finite element saddle point systems arising from the discretization of the Stokes problems to test their performance.

We shall derive and propose several efficient domain decomposition methods for solving the nonlinear inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The highly ill-posed inverse problems are transformed into output least-squares nonlinear and non-convex minimizations with classical Tikhonov regularization. The Levenberg–Marquardt method is applied to transform the non-convex minimizations into convex minimizations, which will be solved by several efficient domain decomposition methods. The methods are completely local and the local minimizers have explicit expressions within the subdomains. Several numerical experiments are presented to show the accuracy and efficiency of the methods; in particular, the convergence seems nearly optimal in the sense that the iteration number of the methods is nearly independent of mesh sizes.

In this paper, a semi-discrete Galerkin finite element method is applied to the two-dimensional diffusive Peterlin viscoelastic model which can describe the unsteady behavior of some incompressible ploymeric fluids. For the derived semi-discrete finite element spatial discretization scheme, the a priori bounds are given that does not rely on the mesh width restriction. Further, with the help of the a priori error bounds of the Stokes and Ritz projections, optimal error estimates for the velocity, the conformation tensor and the pressure are presented, respectively. Finally, in order to implement the proposed semi-discrete numerical scheme, we derive three kinds of fully discrete schemes, e.g., Newton’s iterative scheme, Picard’s iterative scheme and implicit-explicit time-stepping scheme. Finally, several numerical experiments are conducted to confirm our theoretical results.

The stochastic Allen–Cahn equation with multiplicative noise involves the nonlinear drift operator 𝒜⁢(x)=Δ⁢x-(|x|2-1)⁢x{{\mathscr{A}}(x)=\Delta x-(|x|^{2}-1)x}. We use the fact that 𝒜⁢(x)=-𝒥′⁢(x){{\mathscr{A}}(x)=-{\mathcal{J}}^{\prime}(x)} satisfies a weak monotonicity property to deduce uniform bounds in strong norms for solutions of the temporal, as well as of the spatio-temporal discretization of the problem. This weak monotonicity property then allows for the estimate sup1≤j≤J⁡𝔼⁢[∥Xtj-Yj∥𝕃22]≤Cδ⁢(k1-δ+h2)\sup_{1\leq j\leq J}{\mathbb{E}}[\|X_{t_{j}}-Y^{j}\|_{{\mathbb{L}}^{2}}^{2}]% \leq C_{\delta}(k^{1-\delta}+h^{2}) for all small δ>0{\delta>0}, where X is the strong variational solution of the stochastic Allen–Cahn equation, while {Yj:0≤j≤J}{\{Y^{j}:0\leq j\leq J\}} solves a structure preserving finite element based space-time discretization of the problem on a temporal mesh {tj:1≤j≤J}{\{t_{j}:1\leq j\leq J\}} of size k>0{k>0} which covers [0,T]{[0,T]}.

An error estimate for the minimal error method for nonlinear ill-posed problems under general a Hölder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a Hölder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate.