Many geological formations consist of crystalline rock that have very low matrix permeability but allow flow through an interconnected network of fractures. Understanding the flow of groundwater through such rocks is important in considering disposal of radioactive waste in underground repositories. A specific area of interest is the conditioning of fracture transmissivities on measured values of pressure in these formations. While there are existing methods to condition transmissivity fields on transmissivity, pressure and flow measurements for a continuous porous medium, considerably less work has been devoted to conditioning discrete fracture networks. This article presents two new methods for conditioning fracture transmissivities on measured pressures in a discrete fracture network. The first approach adopts a linear approximation when fracture transmissivities are mildly heterogeneous, while the minimisation of a suitable objective function is undertaken when fracture transmissivities are highly heterogeneous. The second conditioning algorithm is a Bayesian method that finds a maximum a posteriori (MAP) estimator which maximises the posterior distribution defined by Bayes’ theorem using information from the prior distribution of fracture transmissivities and observations in the form of measured pressures. The conditioning methods are tested on two separate, large scale test cases that model a potential site for radioactive waste disposal. Results from these test cases are shown and comparisons between the two conditioning methods are made.

We propose a LT (Laplace transformation) method for solving the PIDE (partial integro-differential equation) arising from the financial mathematics. An option model under a jump-diffusion process is given by a PIDE, whose non-local integral term requires huge computational costs. In this work, the PIDE is transformed into a set of complex-valued elliptic problems by taking the Laplace transformation in time variable. Only a small number of Laplace transformed equations are then solved on a suitable choice of contour. Then the time-domain solution can be obtained by taking the Laplace inversion based on the chosen contour. Especially a splitting method is proposed to solve the PIDE, and its solvability and convergence are proved. Numerical results are shown to confirm the efficiency of the proposed method and the parallelizable property.

This paper presents barycentric coordinate interpolation reformulated as bilinear and trilinear mixed finite elements on quadrilateral and hexahedral meshes. The new finite element space is a subspace of H(div). Barycentric coordinate interpolations of discrete vector field with node values are also known as the corner velocity interpolation. The benefit of this velocity interpolation is that it contains the constant vector fields (uniform flow). We provide edge based basis functions ensuring the same interpolation, and show how these basis functions perform as separate velocity elements.

This paper is concerned with solving nonlinear monotone difference schemes of the parabolic type. The monotone Jacobi and monotone Gauss-Seidel methods are constructed. Convergence rates of the methods are compared and estimated. The proposed methods are applied to solving nonlinear singularly perturbed parabolic problems. Uniform convergence of the monotone methods is proved. Numerical experiments complement the theoretical results.

In this paper, we study adaptive finite element approximations in a perturbation framework, which makes use of the existing adaptive finite element analysis of a linear symmetric elliptic problem. We analyze the convergence and complexity of adaptive finite element methods for a class of elliptic partial differential equations when the initial finite element mesh is sufficiently fine. For illustration, we apply the general approach to obtain the convergence and complexity of adaptive finite element methods for a nonsymmetric problem, a nonlinear problem as well as an unbounded coefficient eigenvalue problem.

A new eXtended Finite Element Method based on two-dimensional edge elements is presented and applied to solve the time-harmonic Maxwell equations in domains with cracks. Error analysis is performed and shows the method to be convergent with an order of at least $O(h^{1/2-\eta})$. The implementation of the method is discussed and numerical tests illustrate its performance.

The market pricing of OTC FX options displays both stochastic volatility and stochastic skewness in the risk-neutral distribution governing currency returns. To capture this unique phenomenon Carr and Wu developed a model (SSM) with three dynamical state variables. They then used Fourier methods to value simple European-style options. However, pricing exotic options requires numerical solution of 3D unsteady PIDE with mixed derivatives which is expensive. In this paper to achieve this goal we propose a new splitting technique. Being combined with another method of the authors, which uses pseudo-parabolic PDE instead of PIDE, this reduces the original 3D unsteady problem to a set of 1D unsteady PDEs, thus allowing a significant computational speedup. We demonstrate this technique for single and double barrier options priced using the SSM.

A coupled system, which consists of multiple delayed neural network loops, is proposed and a detailed analysis of the asymptotic behavior of the zero solution is included. The stable regions and all possible bifurcations, which depend on multiple parameters, are given in a geometrical way for several specific cases.