We present an expanded mixed finite element method for solving second-order elliptic partial differential equations on geometrically general domains. For the lowest-order Raviart–Thomas approximating spaces, we use quadrature rules to reduce the method to cell-centered finite differences, possibly enhanced with some face-centered pressures. This substantially reduces the computational complexit...

In this paper, double integrals over an arbitrary quadrilateral are evaluated exploiting finite element method. The physical region is transformed into a standard quadrilateral finite element using the basis functions in local space. Then the standard quadrilateral is subdivided into two triangles, and each triangle is further discretized into 4 n right isosceles triangles, with area 1 2n2, and...

We investigate the use of numerical quadrature in the p-version of the finite element method. We describe a set of minimal conditions that the quadrature rules should satisfy, for various types of elements. Under sufficient assumptions of smoothness, we establish optimality of the asymptotic rate of convergence. Some computational results are presented, which illustrate under what conditions ov...

Abstract. A fully discrete approximation of the linear stochastic wave equation driven by additive noise is presented. A standard finite element method is used for the spatial discretisation and a stochastic trigonometric scheme for the temporal approximation. This explicit time integrator allows for error bounds independent of the space discretisation and thus do not have a step size restricti...

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An analysis is given for a class of nonconforming Lagrange-type finite elements which have been successfully utilized to approximate the solution of a variational problem modeling the deformation of martensitic crystals with microstructure. These elements were first proposed and analyzed in 1992 by Rannacher and Turek for the Stokes equation. Our analysis highlights the features of these elemen...

The explicit finite-difference method for solving variable order fractional space-time wave equation with a nonlinear source term is considered.The concept of variable order fractional derivative is considered in the sense of Caputo.The stability analysis and the truncation error of the method are discussed. To demonstrate the effectiveness of the method, some numerical test examples are presen...

We derive upper and lower a posteriori estimates for the maximum norm error in finite element solutions of monotone semi-linear equations. The estimates hold for Lagrange elements of any fixed order, non-smooth nonlinearities, and take numerical integration into account. The proof hinges on constructing continuous barrier functions by correcting the discrete solution appropriately, and then app...

In order to numerically solve a second-order linear elliptic boundary value problem in a bounded domain, using the finite element method, it is often necessary in practice to violate certain assumptions of the standard variational formulation. Two of these "variational crimes" will be emphasized here and it will be shown that optimal L error estimates still hold. The first "crime" occurs when a...

Abstract. In this paper the numerical approximation of solutions of Liouville-Master Equations for time-dependent distribution functions of Piecewise Deterministic Processes with memory is considered. These equations are linear hyperbolic PDEs with non-constant coefficients, and boundary conditions that depend on integrals over the interior of the integration domain. We construct a finite diffe...