Galerkin Time-stepping Method Research Articles

Discrete Maximal Regularity for the Discontinuous Galerkin Time-Stepping Method without Logarithmic Factor

Discrete Maximal Regularity for the Discontinuous Galerkin Time-Stepping Method without Logarithmic Factor

An hp-version of the discontinuous Galerkin time-stepping method for Volterra integral equations with weakly singular kernels

We develop and analyze an hp-version of the discontinuous Galerkin time-stepping method for linear Volterra integral equations with weakly singular kernels. We derive a priori error bound in the L2-norm that is fully explicit in the local time steps, the local approximation orders, and the local regularity of the exact solutions. For solutions with singular behavior near t=0 caused by the weakly singular kernels, we prove optimal algebraic convergence rates for the h-version of the discontinuous Galerkin approximations on graded meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometrically refined time steps and linearly increasing approximation orders. Numerical experiments are presented to illustrate the theoretical results.

Variational analysis of the discontinuous Galerkin time-stepping method for parabolic equations

Abstract The discontinuous Galerkin (DG) time-stepping method applied to abstract evolution equation of parabolic type is studied using a variational approach. We establish the inf-sup condition or Babuška–Brezzi condition for the DG bilinear form. Then, a nearly best approximation property and a nearly symmetric error estimate are obtained as corollaries. Moreover, the optimal order error estimates under appropriate regularity assumption on the solution are derived as direct applications of the standard interpolation error estimates. Our method of analysis is new for the DG time-stepping method; it differs from previous works by which the method is formulated as the one-step method. We apply our abstract results to the finite element approximation of a second-order parabolic equation with space-time variable coefficient functions in a polyhedral domain, and derive the optimal order error estimates in several norms.

An [formula omitted]-version of the discontinuous Galerkin time-stepping method for nonlinear second-order delay differential equations with vanishing delays

We present and analyze an hp-version of the discontinuous Galerkin (DG) time-stepping method for nonlinear second-order delay differential equations with vanishing delays. We derive a priori error bounds in the L2- and L∞-norm that are fully explicit with respect to the local time steps, the local approximation orders, and the local regularity of the exact solution. We further prove that the hp-DG scheme based on geometrically refined time steps and on linearly increasing approximation orders attains exponential rates of convergence for analytic solutions with start-up singularities. Moreover, we also propose an hp-DG scheme for the state dependent delay differential equations. We illustrate the theoretical results with a series of numerical experiments.

Preconditioners for the Discontinuous Galerkin time-stepping method of arbitrary order

We develop a preconditioner for systems arising from space-time finite element discretizations of parabolic equations. The preconditioner is based on a transformation of the coupled system into block diagonal form and an efficient solution strategy for the arising 2 × 2 blocks. The suggested strategy makes use of an inexact factorization of the Schur complement of these blocks, for which uniform bounds on the condition number can be proven. The main computational effort of the preconditioner lies in solving implicit Euler-like problems, which allows for the usage of efficient standard solvers. Numerical experiments are performed to corroborate our theoretical findings.

Robust and efficient preconditioners for the discontinuous Galerkin time-stepping method

The discontinuous Galerkin time-stepping method has many advantageous properties for solving parabolic equations. However, it requires the solution of a large nonsymmetric system at each time-step. This work develops a fully robust and efficient preconditioning strategy for solving these systems. Drawing on parabolic inf-sup theory, we first construct a left preconditioner that transforms the linear system to a symmetric positive definite problem to be solved by the preconditioned conjugate gradient algorithm. We then prove that the transformed system can be further preconditioned by an ideal block diagonal preconditioner, leading to a condition number bounded by 4 for any time-step size, any approximation order and any positive-definite self-adjoint spatial operators. Numerical experiments demonstrate the low condition numbers and fast convergence of the algorithm for both ideal and approximate preconditioners, and show the feasibility of the high-order solution of large problems.

Mixed Discontinuous Galerkin Time-Stepping Method for Semilinear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to semilinear parabolic optimal control problems, where the discontinuous finite element method of the order r (r ? 0) is used for the time discretization and the Raviart---Thomas mixed finite element method of the order ? (? ? 0) is used for the space discretization. For ? ? 0, r = 0 or 1, we derive a priori error estimates for both the control variable and the state variables. Moveover, we derive a posteriori L2(0,T;L2(Ω)) error estimates for the scalar functions, assuming that only the underlying mesh is static.

Discontinuous Galerkin finite element methods for time-dependent Hamilton–Jacobi–Bellman equations with Cordes coefficients

We propose and analyse a fully discrete discontinuous Galerkin time-stepping method for parabolic Hamilton–Jacobi–Bellman equations with Cordes coefficients. The method is consistent and unconditionally stable on rather general unstructured meshes and time-partitions. Error bounds for both rough and regular solutions in terms of temporal regularity show that the method is arbitrarily high-order with optimal convergence rates with respect to the mesh size, time-interval length and temporal polynomial degree, and possibly suboptimal by an order and a half in the spatial polynomial degree. Numerical experiments on problems with strongly anisotropic diffusion coefficients and early-time singularities demonstrate the accuracy and computational efficiency of the method, with exponential convergence rates achieved under combined hp- and $$\tau q$$ -refinement.

Mixed Discontinuous Galerkin Time-Stepping Method for Linear Parabolic Optimal Control Problems

In this paper, we discuss the mixed discontinuous Galerkin (DG) finite element approximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time discretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We derive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori L 2 (0,T;L 2 ()) error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.

Finite element and DG analysis for neutral-type Volterra integro-differential equations with weakly singular kernels

In this paper we analyze the numerical solution of Volterra integro-differential equations of neutral type with weakly singular kernels. We establish a priori error estimations for the finite-element-method semi-discretization of the given problem by defining a suitable Ritz–Volterra projection operator: here, the key point in the proof is the fact that the definition of the Ritz–Volterra projection operator that is not related to the neutral term. We then discuss the discontinuous Galerkin time-stepping method for the semi-discretized equation, together with a fully discretized form.

Finite element methods for partial Volterra integro-differential equations on two-dimensional unbounded spatial domains

The precise and effective way for solving a partial Volterra integro-differential equation on unbounded spatial domain is to derive artificial boundary conditions for some feasible and prescribed computational domain. Thus the unsolvable problem on unbounded domain is converted to the computational problem on bounded domain, which is often called reduced problem. A feasible numerical method will then be applied to solve the reduced problem with the artificial boundary conditions which often exhibit “nonlocal” properties. In this paper, the finite element method for the spatial space and the recently developed discontinuous Galerkin time-stepping method for the temporal space are considered to solve the reduced problem. The convergence rate, explicitly with those interesting parameters: spatial grid-size h, temporal mesh-size k, the location of the artificial boundary d, and the truncation number M, is obtained. This paper is the first theoretical analysis of the numerical methods for Volterra integro-differential equations on unbounded spatial domains.

Hp-Discontinuous Galerkin Time-Stepping for Volterra Integrodifferential Equations

We present an $hp$-error analysis of the discontinuous Galerkin time-stepping method for Volterra integrodifferential equations with weakly singular kernels. We derive new error bounds that are explicit in the time-steps, the degrees of the approximating polynomials, and the regularity properties of the exact solution. It is then shown that start-up singularities can be resolved at exponential rates of convergence by using geometrically graded time-steps. Our theoretical results are confirmed in a series of numerical tests.

A Posteriori Error Estimates for Discontinuous Galerkin Time-Stepping Method for Optimal Control Problems Governed by Parabolic Equations

In this paper, we examine the discontinuous Galerkin (DG) finite element approximation to convex distributed optimal control problems governed by linear parabolic equations, where the discontinuous finite element method is used for the time discretization and the conforming finite element method is used for the space discretization. We derive a posteriori error estimates for both the state and the control approximation, assuming only that the underlying mesh in space is nondegenerate. For problems with control constraints of obstacle type, which are the kind most frequently met in applications, further improved error estimates are obtained.

Hp-discontinuous Galerkin time-stepping for parabolic problems

We consider the hp-version of the discontinuous Galerkin (DG) time-stepping method for linear parabolic problems with non-symmetric elliptic spatial operators. We derive new analyticity estimates for the exact solutions by means of semigroup techniques. These estimates allow us to show that the hp-DG time-stepping method can resolve start-up singularities at exponential rates of convergence.

An hp a priori error analysis of¶the DG time-stepping method for initial value problems

The Discontinuous Galerkin (DG) time-stepping method for the numerical solution of initial value ODEs is analyzed in the context of the hp-version of the Galerkin method. New a priori error bounds explicit in the time steps and in the approximation orders are derived and it is proved that the DG method gives spectral and exponential accuracy for problems with smooth and analytic time dependence, respectively. It is further shown that temporal singularities can be resolved at exponential rates of convergence if geometrically refined time steps are employed.