A Hybrid Collocation Method for Volterra Integral Equations with Weakly Singular Kernels

The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving ( nonpolynomial) collocation method is known to have only local convergence. To overcome the shortcoming of these well-known methods, we introduce a hybrid collocation method for solving Volterra integral equations of the second kind with weakly singular kernels. In this hybrid method we combine a singularity preserving ( nonpolynomial) collocation method used near the singular point of the derivative of the solution and a graded piecewise polynomial collocation method used for the rest of the domain. We prove the optimal order of global convergence for this method. The convergence analysis of this method is based on a singularity expansion of the exact solution of the equations. We prove that the solutions of such equations can be decomposed into two parts, with one part being a linear combination of some known singular functions which reflect the singularity of the solutions and the other part being a smooth function. A numerical example is presented to demonstrate the effectiveness of the proposed method and to compare it to the graded collocation method

Finite element methods for semilinear elliptic stochastic partial differential equations

We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results

Analysis and Finite Element Approximation of a Coupled, Continuum Pipe-Flow/Darcy Model for Flow in Porous Media with Embedded Conduits

We consider the continuum Darcy/pipe flow model for flows in a porous matrix containing embedded conduits; such coupled flows are present in, e.g., karst aquifers. the mathematical well-posedness of the coupled problem as well as convergence rates of finite element approximation are established in the two-dimensional case. Computational results are also provided. © 2010 Wiley Periodicals, Inc

Numerical analysis of a time discretized method for nonlinear filtering problem with L\'evy process observations

In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is a unnormalized probability density function of the filter solution. Then we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis

Experimental and Computational Validation and Verification of the Stokes-Darcy and Continuum Pipe Flow Models for Karst Aquifers with Dual Porosity Structure

In our previous study, we developed the Stokes-Darcy (SD) model was developed for flow in a karst aquifer with a conduit bedded in matrix, and the Beavers-Joseph (BJ) condition was used to describe the matrix-conduit interface. We also studied the mathematical well-posedness of a coupled continuum pipe flow (CCPF) model as well as convergence rates of its finite element approximation. in this study, to compare the SD model with the CCPF model, we used numerical analyses to validate finite element discretisation methods for the two models. using computational experiments, simulation codes implementing the finite element discretisations are then verified. Further model validation studies are based on the results of laboratory experiments. Comparing the results of computer simulations and experiments, we concluded that the SD model with the Beavers-Joseph interface condition is a valid model for conduit-matrix systems. on the other hand, the CCPF model with the value of the exchange parameter chosen within the range suggested in the literature perhaps does not result in good agreement with experimental observations. We then examined the sensitivity of the CCPF model with respect to the exchange parameter, concluding that, as has previously been noted, the model is highly sensitive for small values of the exchange parameter. However, for larger values, the model becomes less sensitive and, more important, also produces results that are in better agreement with experimental observations. This suggests that the CCPF model may also produce accurate simulation results, if one chooses larger values of the exchange parameter than those suggested in the literature. © 2011 John Wiley & Sons, Ltd