A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu \cite{xu1994novel}, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse mesh $\mathcal{T}_H$ with a low level stochastic collocation (corresponding to the polynomial space $\mathcal{P}_{\boldsymbol{P}}$) and solve linearized equations on a fine mesh $\mathcal{T}_h$ using high level stochastic collocation (corresponding to the polynomial space $\mathcal{P}_{\boldsymbol{p}}$). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with $\mathcal{T}_h$ and $\mathcal{P}_{\boldsymbol{p}}$. The two-level method is computationally more efficient than the standard stochastic collocation method for solving nonlinear problems with random coefficients. Numerical experiments are provided to verify the theoretical results.Comment: 20 pages, 2 figure

Two-grid algorithms for singularly perturbed reaction-diffusion problems on layer adapted meshes

We propose a new two-grid approach based on Bellman-Kalaba quasilinearization and Axelsson-Xu finite element two-grid method for the solution of singularly perturbed reaction-diffusion equations. The algorithms involve solving one inexpensive problem on coarse grid and solving on fine grid one linear problem obtained by quasilinearization of the differential equation about an interpolant of the computed solution on the coarse grid. Different meshes (of Bakhvalov, Shishkin and Vulanovi\'c types) are examined. All the schemes are uniformly convergent with respect to the small parameter. We show theoretically and numerically that the global error of the two-grid method is the same as of the nonlinear problem solved directly on the fine layer-adapted mesh.Comment: 15 pages, 8 figure

Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

We seek discrete approximations to solutions $u:\Omega \to R$ of semilinear elliptic partial differential equations of the form $\Delta u + f_s(u) = 0$, where $f_s$ is a one-parameter family of nonlinear functions and $\Omega$ is a domain in $R^d$. The main achievement of this paper is the approximation of solutions to the PDE on the cube $\Omega=(0,\pi)^3 \subseteq R^3$. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends the authors' work in {\it Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs} (Int. J. of Bifurcation and Chaos, 2009), wherein they combined symmetry analysis with modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our paper {\it An MPI Implementation of a Self-Submitting Parallel Job Queue} (Int. J. of Parallel Prog., 2012).Comment: 37 pages, 25 figure

A new extrapolation cascadic multigrid method for 3D elliptic boundary value problems on rectangular domains

In this paper, we develop a new extrapolation cascadic multigrid (ECMG$_{jcg}$) method, which makes it possible to solve 3D elliptic boundary value problems on rectangular domains of over 100 million unknowns on a desktop computer in minutes. First, by combining Richardson extrapolation and tri-quadratic Serendipity interpolation techniques, we introduce a new extrapolation formula to provide a good initial guess for the iterative solution on the next finer grid, which is a third order approximation to the finite element (FE) solution. And the resulting large sparse linear system from the FE discretization is then solved by the Jacobi-preconditioned Conjugate Gradient (JCG) method. Additionally, instead of performing a fixed number of iterations as cascadic multigrid (CMG) methods, a relative residual stopping criterion is used in iterative solvers, which enables us to obtain conveniently the numerical solution with the desired accuracy. Moreover, a simple Richardson extrapolation is used to cheaply get a fourth order approximate solution on the entire fine grid. Test results are reported to show that ECMG$_{jcg}$ has much better efficiency compared to the classical MG methods. Since the initial guess for the iterative solution is a quite good approximation to the FE solution, numerical results show that only few number of iterations are required on the finest grid for ECMG$_{jcg}$ with an appropriate tolerance of the relative residual to achieve full second order accuracy, which is particularly important when solving large systems of equations and can greatly reduce the computational cost. It should be pointed out that when the tolerance becomes smaller, ECMG$_{jcg}$ still needs only few iterations to obtain fourth order extrapolated solution on each grid, except on the finest grid. Finally, we present the reason why our ECMG algorithms are so highly efficient for solving such problems.Comment: 20 pages, 4 figures, 10 tables; abbreviated abstrac

TGMFE Algorithm Combined with Some Time Second-Order Schemes for Nonlinear Fourth-Order Reaction Diffusion System

In this article, a two-grid mixed finite element (TGMFE) method with some second-order time discrete schemes is developed for numerically solving nonlinear fourth-order reaction diffusion equation. The two-grid MFE method is used to approximate spatial direction, and some second-order $\theta$ schemes formulated at time $t_{k-\theta}$ are considered to discretize the time direction. TGMFE method covers two main steps: a nonlinear MFE system based on the space coarse grid is solved by the iterative algorithm and a coarse solution is arrived at, then a linearized MFE system with fine grid is considered and a TGMFE solution is obtained. Here, the stability and a priori error estimates in $L^2$-norm for both nonlinear Galerkin MFE system and TGMFE scheme are derived. Finally, some convergence results are computed for both nonlinear Galerkin MFE system and TGMFE scheme to verify our theoretical analysis, which show that the convergence rate of the time second-order $\theta$ scheme including Crank-Nicolson scheme and second-order backward difference scheme is close to $2$, and that with the comparison to the computing time of nonlinear Galerkin MFE method, the CPU-time by using TGMFE method can be saved

A Two-Grid Finite Element Approximation for A Nonlinear Time-Fractional Cable Equation

In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Gr\"unwald difference (WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The studied algorithm in this paper mainly covers two steps: First, the numerical solution of nonlinear FE scheme on the coarse grid is solved, Second, based on the solution of initial iteration on the coarse grid, the linearized FE system on the fine grid is solved by using Newton iteration. Here, the stability based on fully discrete two-grid method is derived. Moreover, the a priori estimates with second-order convergence rate in time is proved in detail, which is higher than the L1-approximation result with $O(\tau^{2-\alpha}+\tau^{2-\beta})$. Finally, the numerical results by using the two-grid method and FE method are calculated, respectively, and the CPU-time is compared to verify our theoretical results.Comment: 23 pages, 5 figure

Numerical Computations of Viscous, Incompressible Flow Problems Using a Two-Level Finite Element Method

We consider two-level finite element discretization methods for the stream function formulation of the Navier-Stokes equations. The two-level method consists of solving a small nonlinear system on the coarse mesh, then solving a linear system on the fine mesh. The basic result states that the errors between the coarse and fine meshes are related superlinearly. This paper demonstrates that the two-level method can be implemented to approximate efficiently solutions to the Navier-Stokes equations. Two fluid flow calculations are considered to test problems which have a known solution and the driven cavity problem. Stream function contours are displayed showing the main features of the flow.Comment: 11 pages, 18 figure

Pointwise a posteriori error bounds for blow-up in the semilinear heat equation

This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for the semilinear heat equation with a general local Lipschitz reaction term whose solution may blow-up in finite time. More specifically, conditional a posteriori error bounds are derived in the $L^{\infty}L^{\infty}$ norm for a first order in time, implicit-explicit (IMEX), conforming finite element method in space discretization of the problem. Numerical experiments applied to both blow-up and non blow-up cases highlight the generality of our approach and complement the theoretical results

Acceleration of weak Galerkin methods for the Laplacian eigenvalue problem

Recently, we proposed a weak Galerkin finite element method for the Laplace eigenvalue problem. In this paper, we present two-grid and two-space skills to accelerate the weak Galerkin method. By choosing parameters properly, the two-grid and two-space weak Galerkin method not only doubles the convergence rate, but also maintains the asymptotic lower bounds property of the weak Galerkin method. Some numerical examples are provided to validate our theoretical analysis.Comment: 4 figure2, 20 page

Trust-Region Methods for Nonlinear Elliptic Equations with Radial Basis Functions

We consider the numerical solution of nonlinear elliptic boundary value problems with Kansa's method. We derive analytic formulas for the Jacobian and Hessian of the resulting nonlinear collocation system and exploit them within the framework of the trust-region algorithm. This ansatz is tested on semilinear, quasilinear and fully nonlinear elliptic PDEs (including Plateau's problem, Hele-Shaw flow and the Monge-Amp\`ere equation) with excellent results. The new approach distinctly outperforms previous ones based on linearization or finite-difference Jacobians.Comment: Original research pape