## Abstract

We analyze backward Euler time stepping schemes for a primal DPG formulation of a class of parabolic problems.
Optimal error estimates are shown in a natural norm and in the

## 1 Introduction

In this work we analyze a backward Euler primal DPG time stepping scheme for the parabolic problem

(1.1)

Here,

The discontinuous Petrov–Galerkin method with optimal test functions (DPG) pertains to the class of minimum residual methods and was introduced in a series of papers [5, 6, 10]. It has successfully been applied to elliptic problems, see, e.g., [7, 8] for the Poisson problem and [18, 16] for fourth-order problems. Through the use of optimal test functions, the discrete problem inherits the stability of the continuous problem. This comes in advantageous for problems where robustness is one of the main challenges, e.g., singularly perturbed problems, see [24] for reaction-dominated diffusion problems and [11, 3] for the convection-dominated case. Space-time DPG methods have been studied previously, see, e.g., [9, 13, 12]. For other space-time minimum residual methods we refer to [21, 1, 30, 28]. Approaches employing the DPG methodology for the time discretization of parabolic and hyperbolic initial value problems have recently been investigated, cf. [25, 26]. On the other hand, time-stepping methods for ODEs are frequently employed in combination with standard Galerkin finite element methods in space, cf. the monograph [29] for parabolic equations, but less so with DPG methods. To the best of our knowledge there exist only two works in this direction, dealing with time-stepping and spatial DPG methods for the heat equation, namely [19] and [27].

In [19], a backward Euler method is used to discretize in time, and then the DPG methodology is applied to the ultraweak
variational formulation of the resulting equations. The a priori analysis given there employs the Galerkin projection with respect to these very
equations, and hence a spatial discretization error has to be accounted for in every time step. This gives rise to a theoretical error bound of order
*h* and *k* being the spatial mesh-width and time step, respectively. As numerical experiments from [27] indicate, this asymptotic error bound is not optimal.
The authors of [27] study general θ-schemes (including the backward Euler and the Crank–Nicolson time discretization)
based on the primal DPG method [8] and the ultraweak DPG method [7], and provide an
extensive numerical study and comparison of the different approaches, and it turns out that

Our motivation for the present work is to give a theoretically sound explanation of the optimal convergence rates
seen in the numerical experiments from [27] for a backward Euler primal DPG method.
A DPG approximation is attractive, e.g., for singularly perturbed problems not considered here, cf. [11, 24].
An application for a primal DPG method can be found in the recent work [23].
We will consider general second order linear elliptic spatial differential operators.
For the heat equation (
*elliptic projection operators*, was introduced in [31] and is by now one of the main
tools in the analysis of time-stepping FEMs, as witnessed again in [29].
In the present work we prove optimal error estimates in the context of practical primal DPG methods by the use of an elliptic projection operator.
We follow some ideas from [20] where an elliptic projection in the analysis of time-stepping first-order
system least-squares finite element methods is used in the same spirit.

The remainder of this paper is organized as follows:
In Section 2 we introduce the fully discrete method as well as the necessary notation. We prove stability
of the method and provide quasi-optimality results for the elliptic projection operator.
In Section 3 we use these results to show optimal error estimates in the

## 2 Time-Stepping DPG Formulation

### 2.1 Notation

The notation
*T*, but is independent of involved functions.
We write
*T* is not (at most) linear, we explicitly state the dependence.
Moreover, we use the

We consider a time discretization

where

The trial space of our method will be

equipped with the norm

and the test space will be

equipped with the norm

Here,

where

### Lemma 1.

*One has*

### Proof.

The proof follows by now well-established arguments, see [4] or [17] for problems where the norms depend on parameters.
Note that if we use

Throughout this work we will use that

for all

### 2.2 Primal DPG Formulation

For a simpler notation, we use superindices to indicate time evaluations, i.e., for a time-dependent function
*u* is the exact solution of
the parabolic equation.
We approximate

which admits a unique solution

and integrating by parts one obtains the primal DPG formulation

We introduce some bilinear forms and the right-hand side functional:
For

Therefore, our formulation simply reads

We note that the bilinear form
*a* is not bounded independently of *k*
using the above norms, but rather

where

#### 2.2.1 Fully Discrete Scheme

By

For the discretization of traces we use the space of facewise polynomials denoted by
*p*-th order Raviart–Thomas space.
We consider the spaces

The (discrete) trial-to-test operator is given by

We recall that the inner product is given by

The fully discrete scheme then reads: Given

The next lemma establishes coercivity of

#### Lemma 2.

*Let
*

*If, additionally,
*

*Here,
*

#### Proof.

For the first part we use that

This finishes the proof of the first estimate.

For the second one we use some results established in the literature on fully discrete DPG formulations (practical
DPG), e.g., [22].
Let

Moreover, from [22, proof of Lemma 3.2] we infer that

Thus, using the assumption

Combining this with (2.3) we conclude that

Then

Note that the involved constant only depends on the end time *T* and the coefficients, but is otherwise independent of *k* or *h*.
To see the possible dependence on *T* we note that the triangle and Poincaré inequalities as well as boundedness of the coefficients imply that

This finishes the proof. ∎

It should be noted that the condition
*k* independent estimate for the
trace norm. If this condition is not satisfied then the constant depends on *k*.

#### Theorem 3.

*Problem (2.2) is well posed. In particular, the solutions are stable in the sense that*

#### Proof.

According to Lemma 2,
Problem (2.2) admits unique solutions

This together with the estimate from Lemma 2 shows that

Iterating the arguments concludes the proof. ∎

#### 2.2.2 Remark: Known Results for a Bubnov–Galerkin Method for the Heat Equation

We recall results from [29, Section 1] for a standard Bubnov–Galerkin backward Euler method:
Given an initial solution

#### Proposition 4 ([29, p. 16 and Theorem 1.5]).

*Let u denote the solution of (1.1) and let
*

*If Ω is convex then,*

*The constants
*

*u*and

*T*.

#### 2.2.3 Remark: Primal DPG for the Heat Equation — The Trivial Case

Let us consider the simplest model which is the heat equation where

It is straightforward to see that for

For any

Let

which is the standard Galerkin FEM, see [29, Section 1]. Thus the primal DPG solution component

Note that this remark is true only if we consider

### 2.3 Elliptic Projection-Type Operator

To obtain optimal error estimates, we introduce an elliptic projection.
The idea goes back to [31] to obtain optimal

We define the elliptic projection operator

In the proofs below and in some results we will use the (semi-)norm

Note that from boundedness of

i.e.,

hence, uniform positive definiteness of

The next lemma establishes boundedness and an inf-sup condition of
*V*.
Recall that
*k* and that
*k*.

### Lemma 5.

*The bilinear form
*

*for all
*

*for all
*

### Proof.

We show only the second boundedness estimate, as the first one follows from

Therefore the component

We note that

Standard estimates then show that

Using that

Observe that

This shows boundedness. In order to show the inf-sup condition, we first establish the coercivity estimate

Recall that the optimal test function is characterized by

Setting

We are going to estimate

Combining this with (2.8) yields

Now we are in a position to establish the inf-sup condition

With the boundedness estimates for

Applying the coercivity estimate (2.7) for the second term yields

Combining the latter two estimates shows that

Young’s inequality finishes the proof of the

### Lemma 6.

*Let
*

*If, additionally,
*

### Proof.

The best approximation properties follow from standard arguments (Babuška’s theorem). We give the details only for sake of completeness:
Let

and using (2.5) and the preceding lemma,

Since

The last term is handled as before which concludes the proof. ∎

Combining the latter quasi-best approximation result with standard approximation properties yields:

### Corollary 7.

*Suppose that the components of
*

*If, additionally,
*

## 3 Optimal Error Estimate in Energy Norm

This section is devoted to proving optimal error estimates in the
*n*-th solution of the backward Euler method (2.2).
We make the same assumptions on the regularity of solutions as in [29, Section 1], cf. Proposition 4.

## Theorem 8.

*Let
*

*h*) such that for

*If, additionally,
*

*In both estimates, the dependence on T is exponential.*

## Proof.

With the elliptic projection operator

We may also use the norm

*Step 1.*
By Corollary 7 we have

*Step 2.*
We derive error equations: First, write

Second, by (2.2),

Third, combining both identities and writing

where

Putting the term with

*Step 3.*
We use the test function

*Step 4.*
We estimate the contributions

Second,

Third, recall the definition of the (discrete) optimal test function, i.e.,

which together with the Cauchy–Schwarz inequality shows that

We need to estimate