Interior a priori error estimates in Sobolev norms are derived from interior RitzGalerkin equations which are common to a class of methods used in approximating solutions of second order elliptic boundary value problems. The estimates are valid for a large class of piecewise polynomial subspaces used in practice, which are defined on both uniform and nonuniform meshes. It is shown that the erro...

This paper proposes semi-discrete and fully discrete hybridizable discontinuous Galerkin (HDG) methods for the Burgers' equation in two three dimensions. In spatial discretization, we use piecewise polynomials of degrees $ k \ (k \geq 1), k-1 l (l = k-1; k) to approximate scalar function, flux variable interface trace respectively. full discretization method, apply a backward Euler scheme tempo...

The aim of this paper is to find the numerical solutions of the second order linear and nonlinear differential equations with Dirichlet, Neumann and Robin boundary conditions. We use the Bernoulli polynomials as linear combination to the approximate solutions of 2nd order boundary value problems. Here the Bernoulli polynomials over the interval [0, 1] are chosen as trial functions so that care ...

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate meth...

The approximate solution of optimization and control problems for systems governed by linear, elliptic partial differential equations is considered. Such problems are most often solved using methods based on the application of the Lagrange multiplier rule followed by discretization through, e.g., a Galerkin finite element method. As an alternative, we show how least-squares finite element metho...

This paper is devoted to find the numerical solutions of the fourth order linear and nonlinear differential equations using piecewise continuous and differentiable polynomials, such as Bernstein, Bernoulli and Legendre polynomials with specified boundary conditions. We derive rigorous matrix formulations for solving linear and non-linear fourth order BVP and special care is taken about how the ...

We identify and study an LDG-hybridizable Galerkin method, which is not an LDGmethod, for second-order elliptic problems in several space dimensions with remarkable convergence properties. Unlike all other known discontinuousGalerkinmethods using polynomials of degree k ≥ 0 for both the potential as well as the flux, the order of convergence in L of both unknowns is k + 1. Moreover, both the ap...

This paper deals with a set of three partial differential equations involving time-fractional derivatives and nonlinear diffusion operators. model helps us to understand the HIV spread transmission into patient. First, we prove existence uniqueness weak solutions mathematical model. Then, Galerkin finite element scheme is implemented approximate solution Further, a-priori error bounds convergen...

An initial–boundary value problem for a generalized 2D Schrödinger equation in a rectangular domain is considered. Approximate solutions of the form c1(x1, t)χ1(x1, x2) + · · ·+ cN(x1, t)χN(x1, x2) are treated, where χ1, . . . ,χN are the first N eigenfunctions of a 1D eigenvalue problem in x2 depending parametrically on x1 and c1, . . . , cN are coefficients to be defined; they are of interest...