Numerical Homogenization of Heterogeneous Fractional Laplacians

In this paper, we develop a numerical multiscale method to solve the fractional Laplacian with a heterogeneous diffusion coefficient. When the coefficient is heterogeneous, this adds to the computational costs. Moreover, the fractional Laplacian is a nonlocal operator in its standard form, however the Caffarelli-Silvestre extension allows for a localization of the equations. This adds a complexity of an extra spacial dimension and a singular/degenerate coefficient depending on the fractional order. Using a sub-grid correction method, we correct the basis functions in a natural weighted Sobolev space and show that these corrections are able to be truncated to design a computationally efficient scheme with optimal convergence rates. A key ingredient of this method is the use of quasi-interpolation operators to construct the fine scale spaces. Since the solution of the extended problem on the critical boundary is of main interest, we construct a projective quasi-interpolation that has both $d$ and $d+1$ dimensional averages over subsets in the spirit of the Scott-Zhang operator. We show that this operator satisfies local stability and local approximation properties in weighted Sobolev spaces. We further show that we can obtain a greater rate of convergence for sufficient smooth forces, and utilizing a global $L^2$ projection on the critical boundary. We present some numerical examples, utilizing our projective quasi-interpolation in dimension $2+1$ for analytic and heterogeneous cases to demonstrate the rates and effectiveness of the method

Iterative Operator-Splitting Methods with higher order Time-Integration Methods and Applications for Parabolic Partial Differential Equations

In this paper we design higher order time integrators for systems of stiff ordinary differential equations. We could combine implicit Runge-Kutta- and BDF-methods with iterative operator-splitting methods to obtain higher order methods. The motivation of decoupling each complicate operator in simpler operators with an adapted time-scale allow us to solve more efficiently our problems. We compare our new methods with the higher order Fractional-Stepping Runge-Kutta methods, developed for stiff ordinary differential equations. The benefit will be the individual handling of each operators with adapted standard higher order time-integrators. The methods are applied to convection-diffusion-reaction equations and we could obtain higher order results. Finally we discuss the iterative operator-splitting methods for the applications to multi-physical problems

Benchmark computation of eigenvalues with large defect for non-selfadjoint elliptic differential operators

In this paper we present benchmark problems for non-selfadjoint elliptic eigenvalue problems with large defect and ascent. We describe the derivation of the benchmark problem with a discontinuous coefficient and mixed boundary conditions. Numerical experiments are performed to investigate the convergence of a Galerkin finite element method with respect to the discretization parameters, the regularity of the problem, and the ascent of the eigenvalue. This allows us to verify the sharpness of the theoretical estimates from the literature with respect to these parameters. We provide numerical evidence about the size of the ascent and show that it is important to consider the mean value for the eigenvalue approximation

P 1 finite element methods for an elliptic optimal control problem with pointwise state constraints

We present theoretical and numerical results for two P finite element methods for an elliptic distributed optimal control problem on general polygonal/polyhedral domains with pointwise state constraints.

An a posteriori analysis of C\u3csup\u3e0\u3c/sup\u3e interior penalty methods for the obstacle problem of clamped Kirchhoff plates

We develop an a posteriori analysis of C interior penalty methods for the displacement obstacle problem of clamped Kirchhoff plates. We show that a residual based error estimator originally designed for C interior penalty methods for the boundary value problem of clamped Kirchhoff plates can also be used for the obstacle problem. We obtain reliability and efficiency estimates for the error estimator and introduce an adaptive algorithm based on this error estimator. Numerical results indicate that the performance of the adaptive algorithm is optimal for both quadratic and cubic C interior penalty methods. 0 0