Galerkin Methods For Parabolic Equations Research Articles

Hybridizable discontinuous Galerkin method with mixed-order spaces for non-linear diffusion equations with internal jumps

We formulate a hybridizable discontinuous Galerkin method for parabolic equations with non-linear tensor-valued coefficients and jump conditions (Henry’s law). The analysis of the proposed scheme indicates the optimal convergence order for mildly non-linear problems. The same order is also obtained in our numerical studies for simplified settings. A series of numerical experiments investigate the effect of choosing different order approximation spaces for various unknowns.

Stability and Error Estimates of a Novel Spectral Deferred Correction Time-Marching with Local Discontinuous Galerkin Methods for Parabolic Equations

Abstract In this paper, we discuss the stability and error estimates of the fully discrete schemes for parabolic equations, in which local discontinuous Galerkin methods with generalized alternating numerical fluxes and a novel spectral deferred correction method based on second-order time integration methods are adopted. With the energy techniques, we obtain both the second- and fourth-order spectral deferred correction time-marching with local discontinuous Galerkin spatial discretization are unconditional stable. The optimal error estimates for the corresponding fully discrete scheme are derived by the aid of the generalized Gauss–Radau projection. We extend the analysis to problems with higher even-order derivatives. Numerical examples are displayed to verify our theoretical results.

Error Estimates on Hybridizable Discontinuous Galerkin Methods for Parabolic Equations with Nonlinear Coefficients

HDG method has been widely used as an effective numerical technique to obtain physically relevant solutions for PDE. In a practical setting, PDE comes with nonlinear coefficients. Hence, it is inevitable to consider how to obtain an approximate solution for PDE with nonlinear coefficients. Research on using HDG method for PDE with nonlinear coefficients has been conducted along with results obtained from computer simulations. However, error analysis on HDG method for such settings has been limited. In this research, we give error estimations of the hybridizable discontinuous Galerkin (HDG) method for parabolic equations with nonlinear coefficients. We first review the classical HDG method and define notions that will be used throughout the paper. Then, we will give bounds for our estimates when nonlinear coefficients obey “Lipschitz” condition. We will then prove our main result that the errors for our estimations are bounded.

The $L^2$-Projection and Quasi-Optimality of Galerkin Methods for Parabolic Equations

We consider linear parabolic initial-boundary value problems and analyze Galerkin approximation in space. With the help of the inf-sup theory, we derive quasi-optimality results with respect to norms that arise from the standard weak formulation and from a formulation requiring only integrability in time. Moreover, we reveal that the $H^1$-stability of the $L^2$-projection is not only sufficient but also necessary for these results. As application, we consider conforming finite element approximation in space and derive a priori error bounds in terms of the local meshsize and piecewise regularity. The regularity is the minimal one indicated by approximation theory and matches regularity results for linear parabolic problems.

A New Discontinuous Galerkin Method for Parabolic Equations with Discontinuous Coefficient

In this paper, a new discontinuous Galerkin method is developed for the parabolic equation with jump coefficients satisfying the continuous flow condition. Theoretical analysis shows that this method is L2 stable. When the finite element space consists of interpolative polynomials of degrees k, the convergent rate of the semi-discrete discontinuous Galerkin scheme has an order of . Numerical examples for both 1-dimensional and 2-dimensional problems demonstrate the validity of the new method.

Positivity preserving high-order local discontinuous Galerkin method for parabolic equations with blow-up solutions

In this paper, we apply positivity-preserving local discontinuous Galerkin (LDG) methods to solve parabolic equations with blow-up solutions. This model is commonly used in combustion problems. However, previous numerical methods are mainly based on a second order finite difference method. This is because the positivity-preserving property can hardly be satisfied for high-order ones, leading to incorrect blow-up time and blow-up sets. Recently, we have applied discontinuous Galerkin methods to linear hyperbolic equations involving δ-singularities and obtained good approximations. For nonlinear problems, some special limiters are constructed to capture the singularities precisely. We will continue this approach and study parabolic equations with blow-up solutions. We will construct special limiters to keep the positivity of the numerical approximations. Due to the Dirichlet boundary conditions, we have to modify the numerical fluxes and the limiters used in the schemes. Numerical experiments demonstrate that our schemes can capture the blow-up sets, and high-order approximations yield better numerical blow-up time.

A New Discontinuous Galerkin Method for Parabolic Equations with Discontinuous Coefficient

A New Discontinuous Galerkin Method for Parabolic Equations with Discontinuous Coefficient

Error Estimates for the Discontinuous Galerkin Methods for Parabolic Equations

The classical discontinuous Galerkin method for a general parabolic equation is analyzed. Symmetric error estimates for schemes of arbitrary order are presented. The ideas developed below relax many assumptions required in previous work. For example, different discrete spaces may be used at each time step, and the spatial operator need not be self-adjoint or independent of time. Our error estimates are posed in terms of projections of the exact solution onto the discrete spaces and are valid under the minimal regularity guaranteed by the natural energy estimate. These projections are local and enjoy optimal approximation properties when the solution is sufficiently regular.

Maximum-norm stability and error estimates in Galerkin methods for parabolic equations in one space variable

Maximum-norm stability and error estimates of best approximation and nonsmooth data types are derived for the approximate solution of a parabolic equation in one space variable, using the continuous in time Galerkin method based on piecewise polynomial approximating functions on a quasi-uniform mesh.

The Galerkin method for parabolic equations with operators of local type (in Russian)

The existence and the uniqueness of solution are investigated for the Cauchy problem $u’(t) + A(t) u(t) + (Mu) (t) = f(t), u(0) = u\_0$, with a family of linear operators $A(t)$ and a local typo operator $M$. In the applications $M$ represents, in particular, the retardation of the argument or an integral operator. It is shown that this Cauchy problem produces an isomorphism between some natural spaces of given data and solutions. The convergence of the Galerkin method takes place in strong norms containing derivatives by t. The rate of convergence is characterized by two-sided estimates. The influence of perturbations, arising in the course of calculating the coefficients of the approximate problem as scalar products, on the approximate solutions is given also by two-sided estimates in the same norms in which the convergence is proved.

On the Smoothing Property of the Galerkin Method for Parabolic Equations

We analyze the Galerkin approximation of the general second-order parabolic initial-boundary value problem. For the second-order continuous time method with initial data only in $L^2 $ we prove an $L^2 $ error estimate of order $O({{h^2 } / t})$. Analogous results are shown to hold for the error in negative Sobolev norms and for the time derivative of the error. Our analysis uses only elementary energy techniques and the same techniques are shown to give a simple analysis of the backward Euler time discretization.

Three-Level Galerkin Methods for Parabolic Equations

A priori error estimates are derived for algebraically linear discrete-time Galerkin methods for approximating solutions of nonlinear parabolic initial-boundary value problems. The methods discussed result from applying a Galerkin procedure in the space variables and using three- level finite difference approximations in the time variable.

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu t =??(a(x)? u),x?Ω?? n ,t? (O,T], subject to the nonlinear boundary conditionu v =g(x,t,u),x??Ω,t? (O,T] and the usual initial condition. Optimal order error estimates are derived both inL 2 (Ω) andH 1 (Ω) norms for all methods treated, including several that produce linear computational procedures.

Galerkin Methods for Parabolic Equations

Galerkin-type methods, both continuous and discrete in time, are considered for approximating solutions of linear and nonlinear parabolic problems. Bounds reducing the estimation of the error to questions in approximation theory are derived for the several methods studied. These methods include procedures that lead to linear algebraic equations even for strongly nonlinear problems. A number of computational questions related to these procedures are also discussed.