Projection-difference Method Research Articles

Convergence of the Projection-Difference Method for the Approximate Solution of a Smoothly Solvable Parabolic Equation with a Weighted Integral Condition

Convergence of the Projection-Difference Method for the Approximate Solution of a Smoothly Solvable Parabolic Equation with a Weighted Integral Condition

Root Mean Square Error Estimates for the Projection-Difference Method for the Approximate Solution of a Parabolic Equation with a Periodic Condition for the Solution

Using the projection-difference method, we construct an approximate solution of an abstract linear parabolic equation in a separable Hilbert space with a periodic condition for a solution. We use the Galerkin method for the spatial variables and the implicit Euler discretization for time. We obtain root mean square estimates of the error of approximate solutions that are effective both in time and spatial variables; these estimates imply the convergence of approximate solutions to an exact solution and allow one to find the convergence rate.

О СХОДИМОСТИ И СКОРОСТИ СХОДИМОСТИ ПРОЕКЦИОННО-РАЗНОСТНОГО МЕТОДА ПРИБЛИЖЕННОГО РЕШЕНИЯ ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ С ВЕСОВЫМ ИНТЕГРАЛЬНЫМ УСЛОВИЕМ

In the Hilbert space the abstract linear parabolic equation with nonlocal weight integral condition for the solution is resolved approximately by projectiondifference method using time-implicit Euler’s method. Approximation of the problem by spatial variables is oriented on the finite element method. Errors estimations of approximate solutions, convergence of approximate solution to exact one and orders of rate of convergence are established.

СХОДИМОСТЬ В СИЛЬНЫХ НОРМАХ ПРОЕКЦИОННО-РАЗНОСТНОГО МЕТОДА РЕШЕНИЯ ПАРАБОЛИЧЕСКОГО УРАВНЕНИЯ С ПЕРИОДИЧЕСКИМ УСЛОВИЕМ НА РЕШЕНИЕ

A smooth soluble abstract linear parabolic equation with the periodic condition on the solution is treated in a separable Hilbert space. This problem is solved approximately by a projection-difference method using the Galerkin method in space and the implicit Euler scheme in time. Effective both in time and in space strong-norm error estimates for approximate solutions, which imply convergence of approximate solutions to the exact solution and order of convergence rate depending of the smoothness of the exact solution, are obtained.

Solving a variational parabolic equation with the periodic condition by a projection-difference method with the Crank–Nicolson scheme in time

A solution to a smoothly solvable linear variational parabolic equation with the periodic condition is sought in a separable Hilbert space by an approximate projection-difference method using an arbitrary finite-dimensional subspace in space variables and the Crank–Nicolson scheme in time. Solvability, uniqueness, and effective error estimates for approximate solutions are proven. We establish the convergence of approximate solutions to a solution as well as the convergence rate sharp in space variables and time.

Параболическое уравнение с периодическм условием на решение и проекционно-разностный метод его приближенного решения

Параболическое уравнение с периодическм условием на решение и проекционно-разностный метод его приближенного решения

Asymptotic estimates of a projection-difference method for an operator-differential equation

In the current paper, we study a Petrov-Galerkin method for a Cauchy problem for an operator-differential equation with a leading self-adjoint operator A and a subordinate linear operator K(t) in a Hilbert space. Error estimates for the approximate solutions are obtained. We consider the full equation discretization based on a two-level difference scheme. New asymptotic estimates for the convergence rate of approximate solutions are obtained in uniform topology. The method is applied to the model parabolic problems.

Projection–difference method for controlled Fourier filtering

A projection---difference method is developed for approximating controlled Fourier filtering for quasilinear parabolic functional-differential equations. The method relies on a projection---difference scheme (PDS) for the approximation of the differential problem and derives a O(?1/2?+?h) bound on the rate of convergence of PDS in the weighted energy norm without prior assumptions of additional smoothness of the generalized solutions. The PDS leads to a natural approximation of the objective functional in the optimal Fourier filtering problem. A bound of the same order is obtained for the rate of convergence in the functional of the problems approximating the Fourier filter control problem.

The convergence rate of a projection-difference method for an abstract coupled problem

We consider the Cauchy problem for a system of abstract differential equations in Hilbert spaces that generalizes some coupled thermoelasticity problems. We obtain an a priori energy estimate for the convergence rate of a projection-difference method as applied to the Cauchy problem with an arbitrary choice of projection subspaces.

Projection and projection-difference methods for the solution of the Navier-Stokes equations

The projection and projection-difference methods for the approximate solution of the nonlinear unsteady Navier-Stokes equations in a bounded two-dimensional region are studied. Asymptotic estimates for the convergence rate of the approximate solutions and the time and space derivatives in the uniform topology are obtained.

Asymptotic error estimates of a linearized projection-difference method for a differential equation with a monotone operator

We study a projection-difference method of solving the Cauchy problem for an operatordifferential equation with a selfadjoint leading operator A(t) and a nonlinear monotone subordinate operator K(·) in a Hilbert space. This method leads to a solution of a system of linear algebraic equations at each time level. Error estimates are derived for approximate solutions as well as for fractional powers of the operator A(t). The method is applied to a model parabolic problem.

Error estimates in the projection-difference method for a hyperbolic-parabolic system of abstract differential equations

We consider a Cauchy problem for a hyperbolic-parabolic system of abstract differential equations in a Hilbert space, which generalizes a number of linear coupled thermoelasticity problems. Energy error estimates for a projection-difference method as applied to the Cauchy problem are established without imposing any special conditions on projection subspaces.

Error estimates for projection-difference methods for differential equations with differentiable operators

We study the projection-difference methods for approximate solving the Cauchy problem for operator-differential equations with a leading self-adjoint operator A(t) and a subordinate linear operator K(t), whose definition domain is independent of t. Operators A(t) and K(t) are assumed to be sufficiently smooth. We obtain estimates for the rate of convergence of approximate solutions to the exact solution as well as those for fractional degrees of an operator similar to A(0).

Error estimates for a projection-difference method for a linear differential-operator equation

We study a projection-difference method for approximately solving the Cauchy problem u′(t) + A(t)u(t) + K(t)u(t) = h(t), u(0) = 0 for a linear differential-operator equation in a Hilbert space, where A(t) is a self-adjoint operator and K(t) is an operator subordinate to A(t). Time discretization is based on a three-level difference scheme, and space discretization is carried out by the Galerkin method. Under certain smoothness conditions on the function h(t), we obtain estimates for the convergence rate of the approximate solutions to the exact solution.

On the solution of a Schrödinger type equation by a projection-difference method with an implicit Euler scheme with respect to time

A linear nonstationary Schrodinger type problem in a separable Hilbert space is approximately solved by a projection-difference method. The problem is discretized in space by the Galerkin method using finite-dimensional subspaces of finite-element type, and an implicit Euler scheme is used with respect to time. We establish error estimates uniform with respect to the time grid for the approximate solutions; as to the spatial variables, the estimates are given in the norm of the original space as well as in the energy norm. The estimates considered here not only permit one to prove the convergence of approximate solutions to the exact solution but also give a numerical characterization of the convergence rate.

Error estimate in the projection-difference method for an abstract quasilinear hyperbolic equation with a nonsmooth right-hand side

A three-level scheme of the projection-difference method for an abstract quasilinear hyperbolic equation is analyzed for convergency in a Hilbert space with variable operator coefficients and a non-smooth (only Bochner integrable) right-hand side. An energy error estimate is derived without any special conditions imposed on projection subspaces.

Projection-difference method for a linear operator-differential equation

Projection-difference method for a linear operator-differential equation

Study of convergence of the projection-difference method for hyperbolic equations

We consider the Cauchy problem for an abstract quasilinear hyperbolic equation with variable operator coefficients and a nonsmooth but Bochner integrable free term in a Hilbert space. Under study is the scheme for approximate solution of this problem which is a combination of the Galerkin scheme in space variables and the three-layer difference scheme with time weights. We establish an a priori energy error estimate without any special conditions on the projection subspaces. We give a concrete form of this estimate in the case when discretization in the space variables is carried out by the finite element method (for a partial differential equation) and by the Galerkin method in Mikhlin form.

Error estimates for schemes of the projection-difference method for abstract quasilinear hyperbolic equations

We study the convergence of the three-layer scheme of the projection-difference method for abstract quasilinear hyperbolic equations in Hilbert space. We establish asymptotic energy error estimates for an arbitrary choice of finite-dimensional subspaces in which the approximation problems are solved.

On the Rate of Convergence of Projection-Difference Methods for Smoothly Solvable Parabolic Equations

A linear parabolic problem in a separable Hilbert space is solved approximately by the projection-difference method. The problem is discretized in space by the Galerkin method orientated towards finite-dimensional subspaces of finite-element type and in time by using the implicit Euler method and the modified Crank-Nicolson scheme. We establish uniform (with respect to the time grid) and mean-square (in space) error estimates for the approximate solutions. These estimates characterize the rate of convergence of errors to zero with respect to both the time and space variables.