HighOrder Numerical Methods for 2D Parabolic Problems in Single and Composite Domains
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Abstract
In this work, we discuss and compare three methods for the numerical approximation of constant and variablecoefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summationbyparts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests—with respect to accuracy and convergence—for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.
Keywords
Parabolic problems Interface models Level set Complex geometry Discontinuous solutions SBP–SAT finite difference Difference potentials Spectral approach Finite element method Cut elements Immersed boundary Stabilization Higher order accuracy and convergenceMathematics Subject Classification
65M06 65M12 65M22 65M55 65M60 65M70 35K201 Introduction
Designing methods for the highorder accurate numerical approximation of partial differential equations (PDE) posed on composite domains with interfaces, or on irregular and geometrically complex domains, is crucial in the modeling and analysis of problems from science and engineering. Such problems may arise, for example, in materials science (models for the evolution of grain boundaries in polycrystalline materials), fluid dynamics (the simulation of homogeneous or multiphase fluids), engineering (wave propagation in an irregular medium or a composite medium with different material properties), biology (models of blood flow or the cardiac action potential), etc. The analytic solutions of the underlying PDE may have nonsmooth or even discontinuous features, particularly at material interfaces or at interfaces within a composite medium. Standard numerical techniques involving finitedifference approximations, finiteelement approximation, etc., may fail to produce an accurate approximation near the interface, leading one to consider and develop new techniques.
There is extensive existing work addressing numerical approximation of PDE posed on composite domains with interfaces or irregular domains, for example, the boundary integral method [11, 56], difference potentials method [3, 6, 26, 27, 58, 65], immersed boundary method [30, 42, 61, 74], immersed interface method [2, 46, 47, 49, 69], ghost fluid method [31, 32, 50, 51], the matched interface and boundary method [82, 84, 85, 86], Cartesian grid embedded boundary method [19, 41, 57, 83], multigrid method for elliptic problems with discontinuous coefficients on an arbitrary interface [18], virtual node method [9, 39], Voronoi interface method [35, 36], the finite difference method [8, 10, 24, 75, 78, 79] and finite volume method [22, 34] based on mapped grids, or cut finite element method [13, 14, 15, 37, 38, 71, 76]. Indeed, there have been great advances in numerical methods for the approximation of PDE posed on composite domains with interfaces, or on irregular domains. However, it is still a challenge to design highorder accurate and computationallyefficient methods for PDE posed in these complicated geometries, especially for timedependent problems, problems with variable coefficients, or problems with general boundary/interface conditions.
The aim of this work is to establish benchmark (test) problems for the numerical approximation of parabolic PDE defined in irregular or composite domains. The considered models (Sect. 2) arise in the study of mass or heat diffusion in single or composite materials, or as simplified models in other areas (e.g., biology, materials science, etc.). The formulated test problems (Sect. 4) are intended (a) to be suitable for comparison of highorder accurate numerical methods—and will be used as such in this study—and (b) to be useful in further research. Moreover, the proposed problems include a wide variety of possibilities relevant in applications, which any robust numerical method should resolve accurately, including constant diffusion; timevarying diffusion; high frequency oscillations in the analytical solution; large jumps in diffusion coefficients, solution, and/or flux; etc. For now, we will consider a simplified geometrical setting, with the intent of setting a “baseline” from which further research, or more involved comparisons, might be conducted. Therefore, in Sect. 2 we will introduce two circular geometries, which are defined either explicitly, or implicitly via a level set function.
In Sect. 3, we briefly introduce the numerical methods we will consider in this work, i.e., second and fourthorder versions of (i) the Cut Finite Element Method (cutFEM); (ii) the Difference Potentials Method (DPM), with Finite Difference approximation as the underlying discretization in the current work; and (iii) the summationbyparts Finite Difference Method combined with the simultaneous approximation term technique (SBP–SAT–FD). These three methods are all modern numerical methods which may be designed for problems in irregular or composite domains, allowing for highorder accurate numerical approximation, even at points close to irregular interfaces or boundaries. We will apply each method to the formulated benchmark problems, and compare results. From the comparisons, we expect to learn what further developments of the methods at hand would be most important.
To resolve geometrical features of irregular domains, both cutFEM and DPM use a Cartesian grid on top of the domain, which need not conform with boundaries or interfaces. These types of methods are often characterized as “immersed” or “embedded”. In the finite difference framework, embedded methods for parabolic problems are developed in [1, 23]. For comparison with cutFEM and DPM, however, in this paper we use a finite difference method based on a conforming approach. The finite difference operators we use satisfy a summationbyparts principle. Then, in combination with the SAT method to weakly impose boundary and interface conditions, an energy estimate of the semidiscretization can be derived to ensure stability. In addition, we use curvilinear grids and transfinite interpolation to resolve complex geometries.
For recent work on SBP–SAT–FD for wave equations in composite domains, see [10, 17, 75, 78], and the two review papers [21, 73]; for recent work in DPM for elliptic/parabolic problems in composite domains with interface defined explicitly, see [3, 4, 5, 6, 26, 27, 28, 58, 59, 65]; and for recent work in cutFEM, see [13, 14, 15, 37, 53, 71].
2 Statement of Problem
In this section, we describe two diffusion problems, which will be the setting for our proposed benchmark (test) problems in Sect. 4. (Recall from Sect. 1 that these models arise, for example, in the study of mass or heat diffusion.) For brevity, in the following discussion, we denote \(u := u(x, y, t)\) and \(u_s := u_s(x, y, t)\), with \(s = 1, 2\).
2.1 The Single Domain Problem
2.2 The Composite Domain Problem
Remark 1
We consider the circular geometries depicted in Fig. 1 as the geometrical setting for our proposed benchmark problems in this work. In applications (Sect. 1), other geometries will likely be considered, some much more complicated than Fig. 1. While our methods can handle more complicated geometry, this is (to the best of our knowledge) the first work looking to establish benchmarks—and compare numerical methods—for parabolic interface problems (3–9). As such, we think that the geometries in Fig. 1 are a good “baseline”—without all the added complexities that more complicated geometries might produce—from which further research, or more involved comparisons, might be done.
To be more specific, we aim to define a simple set of test problems that can be easily implemented and tested for any numerical scheme of interest. With circular domains, it suffices for us to compare/contrast performance of the numerical methods on a simple geometry with smooth boundary versus on a composite domain with fixed interface (explicit or implicit). The approximation of the solution to such compositedomain problems are already challenging for any numerical methods, since (i) the solution may fail to be smooth (or may be discontinuous) at the interface, and (ii) there may be discontinuous material coefficients (\(\lambda _1 \not = \lambda _2\)).
Remark 2
For both the single and composite domain problems, we could also consider other boundary conditions, e.g., a Neumann boundary condition as in [6, 13], etc.
3 Overview of Numerical Methods
3.1 CutFEM
In this section, we give a brief presentation of the cutFEM method. For a more detailed presentation of cutFEM, see, for example, [13, 14, 53].
Let \(\varOmega _s\) be covered by a structured triangulation, \(\mathcal {T}_{s}\), so that each element \(T\in \mathcal {T}_{s}\) has some part inside of \(\varOmega _s\); see Fig. 2a and b. Here, \(s = 1,2\) is an index for the composite domain problem (3–9), which will be omitted when referring to the single domain problem (1, 2). (For the latter, note that \(\mathcal {T}\) covers \(\varOmega \).) Typically \(\mathcal {T}_1\) and \(\mathcal {T}_2\) would be created from a larger mesh by removing some of the cells. Further, let \(\mathcal {T}_\varGamma =\{T\in \mathcal {T}: T \cap \varGamma \ne \emptyset \}\) be the set of intersected elements; see Fig. 2c. In the following, we shall use \(\varGamma \) both for the immersed boundary of the single domain problem and for the immersed interface of the composite domain problem, in order to make the connection to the set \(\mathcal {T}_\varGamma \) clearer.
 1.
For both (3) and (4), multiply the equation for \({u}_{s}\) with a test function \({v}_{s} \in V_h^s\), and then integrate by parts;
 2.
Add terms consistent with the interface and boundary conditions; and
 3.
Add stabilization terms \(j_1\) and \(j_2\) over \(\mathcal {F}_1\) and \(\mathcal {F}_2\), respectively.
In order to use cutFEM, one needs a way to perform integration over the intersected elements \(\mathcal {T}_\varGamma \). For example, with the interface problem, on each element \(K \in \mathcal {T}_\varGamma \), we need a quadrature rule for the \(K\cap \varOmega _1\), \(K\cap \varOmega _2\) and \(K\cap \varGamma \). For the numerical tests in this work (Sect. 4), we represent the geometry by a level set function, and compute highorder accurate quadrature rules with the algorithm from [68].
Remark 3
Optimal (secondorder) convergence was rigorously proven for cutFEM applied to the Poisson problem in [14]. As far as we know, there is no rigorous proof of higherorder convergence for cutFEM, though such a proof would likely be similar to the secondorder case.
3.2 DPM
We continue in this section with a brief introduction to the Difference Potentials Method (DPM), which was originally proposed by Ryaben’kii (see [64, 65, 66], and see [29, 33] for papers in his honor). Our aim is to consider the numerical approximation of PDEs on arbitrary, smooth geometries (defined either explicitly or implicitly) using the DPM together with standard, finitedifference discretizations of (1) or (3, 4) on uniform, Cartesian grids, which need not conform with boundaries or interfaces. To this end, we work with highorder methods for interface problems based on Difference Potentials, which were originally developed in [67] and [3, 4, 5, 6, 26, 28]. We also introduce new developments here for handling implicitlydefined geometries. (The reader can consult [65] for the general theory of the Difference Potentials Method.)
Broadly, the main idea of the DPM is to reduce uniquely solvable and wellposed boundary value problems in a domain \(\varOmega \) to pseudodifferential Boundary Equations with Projections (BEP) on the boundary of \(\varOmega \). First, we introduce a computationally simple auxiliary domain as part of the method. The original domain is embedded into the auxiliary domain, which is then discretized using a uniform Cartesian grid. Next, we define a Difference Potentials operator via the solution of a simple Auxiliary Problem (defined on the auxilairy domain), and construct the discrete, pseudodifferential Boundary Equations with Projections (BEP) at grid points near the continuous boundary or interface \(\varGamma \). (This set of grid points is called the discrete grid boundary.) Once constructed, the BEP are then solved together with the boundary/interface conditions to obtain the value of the solution at the discrete grid boundary. Lastly, using these reconstructed values of the solution at the discrete grid boundary, the approximation to the solution in the domain \(\varOmega \) is obtained through the discrete, generalized Green’s formula.
Mathematically, the DPM is a discrete analog of the method of Calderón’s potentials in the theory of partial differential equations. The DPM, however, does not require explicit knowledge of Green’s functions. Although we use an Auxiliary Problem (AP) discretized by finite differences, the DPM is not limited to this choice of spatial discretization. Indeed, numerical methods based on the idea of Difference Potentials can be designed with whichever choice of spatial discretization is most natural for the problem at hand (e.g., see [25]).
Practically, the main computational complexity of the DPM reduces to the required solutions of the AP, which can be done very efficiently using fast, standard \(\mathcal {O}(N \log N)\) solvers. Moreover, in general the DPM can be applied to problems with general boundary or interface conditions, with no change to the discretization of the PDE.
Let us now briefly introduce the DPM for the numerical approximation of parabolic interface models (3–9). First, we must introduce the pointsets that will be used throughout the DPM. (Note that the main construction of the method below applies to the single domain problem (1, 2), after omitting the index s and replacing interface conditions with boundary conditions; see [6].)
Let \(\varOmega _s\) (\(s=1,2\)) be embedded in a rectangular auxiliary domain \(\varOmega _{s}^0\). Introduce a uniform, Cartesian grid denoted \(M_s^0\) on \(\varOmega _s^0\), with gridspacing \(h_s\). Let \(M_s^+ = M_s^0 \cap \varOmega _s\) denote the grid points inside each subdomain \(\varOmega _s\), and \(M_s^ = M_s^0 {\setminus } M_s^+\) the grid points outside each subdomain \(\varOmega _s\). Note that the auxiliary domains \(\varOmega _1^0\), \(\varOmega _2^0\) and auxiliary grids \(M_1^0\), \(M_2^0\) need not agree, and indeed may be selected completely independently, given considerations regarding accuracy, adaptivity, or efficiency.
The choices of discretization (33) in each subdomain need not be the same. As in [3, 6], one could choose a second and fourthorder discretization on \(M_1^+\) and \(M_2^+\), respectively, given considerations about accuracy, adaptivity, expected regularity of the analytical solution in each domain, etc.
Next, we define the discrete Auxiliary Problem, which plays a central role in the construction of the Difference Potentials operator, the resulting Boundary Equations with Projection at the discrete grid boundary, and in the numerical approximation of the solution via the discrete, generalized Green’s formula.
Definition 1
Remark 4
For a given righthand side \(q_s^{i + 1}\), the solution of the discrete AP (34, 35) defines a discrete Green’s operator \(G_{\Delta t, h}^s q_s^{i + 1}\). The choice of boundary conditions (35) will affect the resulting grid function \(G_{\Delta t, h}^s q_s^{i + 1}\), and thus the Boundary Equations with Projection defined below. However, the choice of boundary conditions (35) in the AP will not affect the numerical approximation of (3–9), so long as the discrete AP is uniquely solvable and wellposed.
Let us also introduce a linear space \(\mathbf {V}_{\gamma _s}\) of all grid functions denoted \(v_{\gamma _s}^{i + 1}\), which are defined on \(\gamma _s\) and extended by zero to the other points of \(N_s^0\). These grid functions are referred to as discrete densities on \(\gamma _s\).
Definition 2
Note that \(P_{N_s^+}^{i + 1} : \mathbf {V}_{\gamma _s} \rightarrow N_s^+\) is a linear operator on the space \(\mathbf {V}_{\gamma _s}\) of densities. Moreover, the coefficients of \(P_{N_s^+}^{i + 1}\) can be computed by solving the AP (Definition 1) with the appropriate density \(v_{\gamma _s}^{i+1}\) defined at the points \((x_{j}, y_{k}) \in \gamma _s\).
Definition 3
(The Trace operator.) Given a grid function \(v_s^{i + 1}\), we denote by \({\text {Tr}}_{\gamma _s} [v_s^{i + 1}]\) the Trace (or Restriction) from \(N_s^+\) to \(\gamma _s\).
Moreover, for a given density \(v_{\gamma _s}^{i + 1}\), denote the trace of the Difference Potential of \(v_{\gamma _s}^{i + 1}\) by \(P_{\gamma _s} v_{\gamma _s}^{i + 1}\). In other words, \(P_{\gamma _s} v_{\gamma _s}^{i + 1} = {\text {Tr}}_{\gamma _s} [P_{N_s^+}^{i + 1} v_{\gamma _s}^{i + 1}]\).
Now we can state the central theorem of the Difference Potentials Method that will allow us to reformulate the finitedifference Eq. (33) on \(M_s^+\) (without imposing any boundary or interface conditions yet) into the equivalent Boundary Equations with Projections on \(\gamma _s\).
Theorem 1
Proof
See [65] for the general theory of DPM (including the proof for general elliptic PDE), or one of [3, 5, 6] for the proof in the case of parabolic interface problems. \(\square \)
Remark 5
A given density \(v_{\gamma _s}^{i+1}\) is the trace of some solution of the fullydiscrete finitedifference Eq. (33) if and only if it is a solution of the BEP.
However, since boundary or interface conditions have not yet been imposed, the BEP will have infinitely many solutions \(u_{\gamma _s}^{i+1}\). As originally disucssed in [3, 4, 5, 6, 26, 28], in this work we consider the following approach in order to find a unique solution of the BEP.
At each time level \(t^{i + 1}\), one can approximate the solution of (3–9) at the discrete grid boundary \(\gamma _s\), using the Cauchy data of (3–9) on the continuous interface \(\varGamma \), up to the desired second or fourthorder accuracy. (By Cauchy data, we mean the trace of the solution of (3–9), together with the trace of its normal derivative, on \(\varGamma \).) Below, we will define an Extension Operator which will extend the Cauchy data of (3–9) from \(\varGamma \) to \(\gamma _s\).
As we will see, the Extension Operator in this work depends only on the given parabolic interface model. Moreover, we will use a finitedimensional, spectral representation for the Cauchy data of (3–9) on \(\varGamma \). Then, we will use the Extension Operator, together with the BEP (38) and the interface conditions (8, 9), to obtain a linear system of equations for the coefficients of the finitedimensional, spectral representation. Hence, the derived BEP will be solved for the unknown coefficients of the Cauchy data. Using this obtained Cauchy data, we will construct the approximation of (3–9) using the Extension Operator, together with the discrete, generalized Green’s formula.
Now we are ready to define the Extension Operator that extends the Cauchy data of (3–9) from \(\varGamma \) to \(\gamma _s\).
Definition 4
Remark 6
It should be also possible to relax regularity assumption on the domain under consideration. For example, one can consider piecewisesmooth, locallysupported basis functions (defined on \(\varGamma \)) as the part of the Extension Operator. For example, [52] use this approach to design a highorder accurate numerical method for the Helmholtz equation, in a geometry with a reentrant corner. Furthermore, [80, 81] combine the DPM together with the XFEM, and design a DPM for linear elasticity in a nonLipschitz domain (with a cut).
Next, in “Appendix 8.3”, we derive a linear system for the coefficients \((c_{1, \nu }^{s, i+1})_{\nu =1}^{\mathcal {N}^0}\) and \((c_{2, \nu }^{s, i+1})_{\nu =1}^{\mathcal {N}^1}\), by combining the interface conditions (8, 9), the BEP (38), the Extension Operator (39–41), and the spectral discretization (46). Then, the numerical approximation \(u_s^{i+1} \approx u_s(x_j, y_k, t^{i + 1})\) of (3–9) at all gridpoints \((x_j, y_k) \in N_s^+\) follows directly from the discrete, generalized Green’s formula, which we state now.
Definition 5
In this work, we also propose a novel feature of DPM, extending the method originally developed in [67] and [3, 4, 5, 6, 26, 28] to the composite domain problem (3–9) with implicitlydefined geometry. The primary difference between Difference Potentials Methods on explicitlydefined versus implicitlydefined composite domains is in the approximation of the interface \(\varGamma \), which must be done accurately and efficiently, in order to maintain the desired second or fourthorder accuracy.
The main idea of DPMbased methods for implicitlydefined geometry is to seek an accurate and efficient explicit parameterization of the implicit boundary/interface. First, we represent the geometry implicitly via a level set function F(x, y) on \(M^0\). Then we construct a local interpolant \(\tilde{F}(x, y)\) of F(x, y) on a subset of \(M^0\) near the continuous interface \(\varGamma \). Next, we parameterize \(\varGamma \) by arclength using numerical quadrature. With this parameterization, we (i) compute the Fourier series expansion from initial conditions for the Cauchy data \(\mathfrak {u}_{s, \varGamma }^{i + 1}\) on the implicit interface \(\varGamma \), and (ii) construct the extension operators (Definition 4) with \(p=2\) or \(p=4\).
Conjecture 1
(Highorder accuracy of the DPM with implicit geometry) Due to the second or fourthorder accuracy (in both space and time) of the underlying discretization (33), the extension operator (42) with \(p=2\) or \(p=4\), and the established error estimates and convergence results for the DPM for general linear elliptic boundary value problems on smooth domains (presented in [33, 62, 63, 65]), we expect second and fourthorder accuracy in the maximum norm for the error in the computed solution (59 or 60) for both the single and composite domain parabolic problems.
Remark 7
Indeed, in the numerical results (Sect. 5) we see that the computed solution (47) at every time level \(t^{i+1}\) has accuracy \(\mathcal {O}(h^2 + \Delta t^2)\) for the secondorder method, and \(\mathcal {O}(h^4 + \Delta t^4)\) for the fourthorder method, for both the single and composite domain problems, with explicit or implicit geometry. See [3, 6, 27, 67] for more details and numerical tests involving explicit (circular and elliptical) geometries.

Step 1 Introduce a computationally simple Auxiliary Domain \(\varOmega _s^0\) (\(s = 1,2\)) and formulate the Auxiliary Problem (AP; Definition 1).

Step 2 At each time step \(t^{i+1}\), compute the Particular Solution \(u_{s}^{i+1} = G^{i+1}_{\Delta t, h} F^{i+1}_{s}\), \((x_j,y_k)\in N_s^+\), using the AP with the righthand side (36).

Step 3 Construct the matrix in the boundary Eq. (79) (discussed in “Appendix 8.3”), derived from the Boundary Equation with Projection (BEP) (38), via several solutions of the AP. (When the diffusion coefficients \(\lambda _s\) are constant, this is done once, as a preprocessing step before the first time step.)

Step 4 Solve boundary Eq. (79) and compute the approximation of the density \(u^{i+1}_{\gamma _s}\), by applying the Extension Operator (42) to the solution of (79).

Step 5 Construct the Difference Potentials \(P_{N_s^+\gamma _s} u^{i+1}_{\gamma _s}\) of the density \(u_{\gamma _s}^{i + 1}\), using the AP with the righthand side (37).

Step 6 Compute the numerical approximation \(u_s^{i+1} \approx u_s(x_j, y_k, t^{i + 1})\) of the PDE (3–9) using the discrete, generalized Green’s formula (47).
3.3 SBP–SAT–FD
We continue in this section with a brief presentation of SBP–SAT–FD, for solving the parabolic problems presented in Sect. 2. For more detailed discussions of the SBP–SAT–FD method, we refer the reader to two review papers [21, 73].
The SBP–SAT–FD method was originally used on Cartesian grids. To resolve complex geometries, we consider a grid mapping approach by transfinite interpolation [43]. A smooth mapping requires that the physical domain is a quadrilateral, possibly with smooth, curved sides. If the physical domain does not have the desired shape, we then partition the physical domain into subdomains, so that each subdomain can be mapped smoothly to the reference domain. As an example, the single domain of equation (1, 2), shown in Figure 5a, is divided into five subdomains. The five subdomains consist of one square subdomain, and four identical quadrilateral subdomains (modulo rotation by \(\pi /2\)) with curved sides. Similarly, the composite domain of equation (3–9) is divided into nine subdomains, as shown in Fig. 5b. Suitable interface conditions are imposed to patch the subdomains together.
Although the sidelength of the centered square is arbitrary (as long as the square is strictly inside the circle), its size and position have a significant impact on the quality of the curvilinear grid. In a highquality mesh, the elements should not be skewed too much, and the sizes of the elements should be nearly uniform. In practice, it is usually difficult to know a priori the optimal way of domain division.
The SBP operators \(D_1\) were first constructed in [45] and later revisited in [72]. The SBP norm H can be diagonal or nondiagonal. While nondiagonal norm SBP operators have a better accuracy property than diagonal norm SBP operators, when terms with variable coefficients are present in the equation, a stability proof is only possible with diagonal norm SBP operators. Therefore, we use diagonal norm SBP operators in this paper.
For a second derivative with variable coefficients, the SBP operators \(D^{(b)}_2\) were constructed in [54]. We remark that applying \(D_1\) twice also approximates a second derivative, but is less accurate and more computationally expensive than \(D^{(b)}_2\).
Due to the choice of centered difference stencils at interior grid points, the order of accuracy of the SBP operators is even at these points, and is often denoted by 2p. To fulfill the SBP property, at a few grid points near boundaries, the order of accuracy is reduced to p for diagonal norm operators. This detail notwithstanding, such a scheme is often referred to as \(2p^{th}\)order accurate. In fact, for the second and fourthorder SBP–SAT–FD schemes used in this paper to solve parabolic problems, we can expect a second and fourthorder overall convergence rate, respectively [77].
An SBP operator only approximates a derivative. When imposing boundary and interface conditions, it is important that the SBP property is preserved and an energy estimate is obtained. For this reason, we consider the SAT method [16], where penalty terms are added to the semidiscretization, imposing the boundary and interface conditions weakly. This bears similarities with the Nitsche finite element method [60] and the discontinuous Galerkin method [40].
In [10], SBP–SAT–FD methods are discussed for the onedimensional heat equation with constant coefficients, both in a single domain and a composite domain. In theory, these schemes can also be generalized to solve Eq. (49), but are different from the ones used in this paper.
4 Test Problems
In this section, we first list the test problems that we will consider (in Sect. 4.1), and then briefly motivate and discuss these choices (in Sect. 4.2). The tests we propose are “manufactured solutions”, in the sense that we state an exact solution u(x, y, t) or \((u_1(x, y, t), \, u_2(x, y, t))\) and a diffusion coefficient \(\lambda (t)\) or \((\lambda _1, \lambda _2)\). From (1, 2) (for the single domain problem) or (3–9) (for the composite domain problem) we compute the (i) righthand side, (ii) initial conditions, (iii) boundary condition, and (iv) functions \((\mu _1(x, y, t), \, \mu _2(x, y, t))\) for the interface/matching conditions. Then, (i–iv), together with the diffusion coefficient, serve as the inputs for our numerical methods.
4.1 List of Test Problems
 1.Singledomain, with an explicitlydefined boundary for DPM and SBP–SAT–FD, or an implicitlydefined boundary for cutFEM.
 (a)
 (b)Timevarying diffusion (Test Problem 3A; TP–3A): Same as TP–1A, but with diffusion coefficient
 (a)
 2.Compositedomain, with an explicitlydefined interface (for DPM and SBP–SAT–FD) or implicitlydefined interface (for cutFEM and DPM). Consider the PDE (3–9), with \(\varOmega = [2, 2] \times [2, 2]\), \(\varOmega _2 = \{ (x, y) \in \mathbb {R}^2 : x^2 + y^2 \le 1 \}\), \(\varOmega _1 = \varOmega {\setminus } \varOmega _2\), \(\varGamma = \{ (x, y) \in \mathbb {R}^2 : x^2 + y^2 = 1 \}\), and the final time \(T = 1.0\).
 (a)
 (b)Highfrequency oscillations (Test Problem 2B; TP–2B): A modified version of the test adapted from [6]. Let \((\lambda _1, \lambda _2) = (10, 1)\), and
 (c)Large contrast in diffusion coefficients, and large jumps in both solution and flux at interface (Test Problem 2C; TP–2C): Let \((\lambda _1, \lambda _2) = (1000, 1)\), and
 (a)
4.2 Motivation of the Chosen Test Problems
Test Problem 1A (TP–1A) involves a highdegree polynomial, with total degree of 17. This is a rather straightforward test problem, which allows us to establish a good “baseline” with which to compare each method. The choice of high degree ensures that there will be no cancellation of local truncation error, so that we should see—at most—second or fourthorder convergence for the given methods, barring some type of superconvergence. Next, (TP–3A) adds on (incrementally) the complication of timevarying diffusion.
Likewise, (TP–2A) offers a straightforward “baseline” with which to consider the interface problem: The test problem is piecewisesmooth, and the geometry is simplified (see Remark 1). However, there is a jump in both the analytical solution and its flux, which requires a welldesigned numerical method to accurately approximate. Moreover, (TP–2A) was first proposed in [48] (see also [6]), and is a good comparison with the immersed interface method therein.
Then, (TP–2B) adds additional challenges onto (TP–2A) in the form of much higherfrequency oscillations; while (TP–2C) adds onto (TP–2A) in the form of both (i) large contrast in diffusion, and (ii) large jumps in the analytical solution and its flux.
5 Numerical Results
5.1 Time Discretization
5.2 Measure for Comparison
5.3 Convergence Results
In the following tables and figures, we state the number of degrees of freedom in the grid, maximum error (59, 60 for the single and compositedomain problems, respectively), and an estimate of the rate of convergence.
Convergence in the maximum norm (59), for the second and fourthorder versions of each method, applied to Test Problem 1A (TP–1A), with diffusion coefficient \(\lambda = 1\), and timestep \(\Delta t = 0.5 h\)
DOF  E: CUT2  Rate  DOF  E: CUT4  Rate 

9944  4.9327E−5  –  10,276  3.1799E−6  – 
40,072  1.3798E−5  1.80  39,613  1.9848E−7  4.00 
159,912  3.7114E−6  1.88  159,700  1.3330E−8  3.82 
DOF  E: DPM2  Rate  DOF  E: DPM4  Rate 

10,000  1.7105E−5  –  10,000  2.4782E−6  – 
40,000  4.1980E−06  2.03  40,000  5.9672E−8  5.38 
160,000  1.0135E−06  2.05  160,000  1.7396E−9  5.10 
DOF  E: SBP2  Rate  DOF  E: SBP4  Rate 

9861  1.5328E−5  –  9861  2.0636E−6  – 
40,365  3.6210E−6  2.08  40,365  1.3083E−7  3.98 
163,317  8.8008E−7  2.04  163,317  8.1180E−9  4.01 
Convergence in the maximum norm (59), for the second and fourthorder versions of each method, applied to Test Problem 3A (TP–3A), with diffusion coefficient \(\lambda (t) = 1.1 + \sin (\pi t)\), and timestep \(\Delta t = 0.5h\)
DOF  E: CUT2  Rate  DOF  E: CUT4  Rate 

9944  4.9605E−5  –  10,276  3.0791E−6  – 
40,072  1.3851E−5  1.80  39,613  1.9435E−7  3.99 
159,912  3.7176E−6  1.89  159,700  1.3161E−8  3.81 
DOF  E: DPM2  Rate  DOF  E: DPM4  Rate 

10,000  1.7721E−5  –  10,000  2.3422E−6  – 
40,000  4.3619E−6  2.02  40,000  5.7588E−8  5.35 
160,000  1.0526E−6  2.05  160,000  1.8398E−9  4.97 
DOF  E: SBP2  Rate  DOF  E: SBP4  Rate 

9861  1.5665E−5  –  9861  1.8858E−6  – 
40,365  3.6965E−6  2.08  40,365  1.1949E−7  3.98 
163,317  8.9731E−7  2.04  163,317  7.4149E−9  4.01 
Convergence in the maximum norm (60), for the second and fourthorder versions of each method, applied to Test Problem 2A (TP–2A), with diffusion coefficients \((\lambda _1, \lambda _2) = (10, 1)\), and timestep \(\Delta t = 0.5h\). (DPM2I/DPM4I refers to the extension of the DPM method, to consider implicit geometry.)
DOF  E: CUT2  Rate  DOF  E: CUT4  Rate 

9988  1.0933E−3  –  10,129  2.2215E−6  – 
39,988  2.7169E−4  1.97  39,952  1.3254E−07  3.98 
159,988  7.2092E−05  1.89  160,729  8.1985E−09  3.93 
DOF  E: DPM2  Rate  DOF  E: DPM4  Rate 

10,000  3.6380E−5  –  10,000  7.7484E−9  – 
40,000  8.8360E−6  2.04  40,000  4.5617E−10  4.09 
160000  2.1331E−6  2.05  160,000  2.6398E−11  4.11 
DOF  E: DPM2I  Rate  DOF  E: DPM4I  Rate 

10,000  3.6381E−5  —  10,000  7.7484E−9  – 
40,000  8.8360E−6  2.04  40,000  4.5617E−10  4.09 
160,000  2.1331E−6  2.05  160,000  2.6396E−11  4.11 
DOF  E: SBP2  Rate  DOF  E: SBP4  Rate 

10,537  4.7387E−4  –  10,537  3.4655E−5  – 
40,905  1.2049E−4  1.98  40,905  4.3052E−6  3.01 
161,161  3.0267E−5  1.99  161,161  5.3535E−7  3.01 
Convergence in the maximum norm (60), for the second and fourthorder versions of each method, applied to Test Problem 2B (TP–2B), with diffusion coefficients \((\lambda _1, \lambda _2) = (10, 1)\), and timestep \(\Delta t = 0.5h\). (DPM2I/DPM4I refers to the extension of the DPM method, to consider implicit geometry.)
DOF  E: CUT2  Rate  DOF  E: CUT4  Rate 

9988  2.4855E−1  –  10,129  4.7064E−1  – 
39,988  5.6850E−2  2.08  39,952  3.6816E−2  3.60 
159,988  1.2346E−2  2.18  160,729  2.2361E−3  3.95 
DOF  E: DPM2  Rate  DOF  E: DPM4  Rate 

10,000  7.1899E−2  –  10,000  7.3065E−3  – 
40,000  1.7868E−2  2.01  40,000  6.0014E−4  3.61 
160,000  4.4952E−3  1.99  160,000  3.3086E−5  4.18 
DOF  E: DPM2I  Rate  DOF  E: DPM4I  Rate 

10,000  7.1899E−2  –  10,000  7.3065E−3  – 
40,000  1.7868E−2  2.01  40,000  6.0014E−4  3.61 
160,000  4.4952E−3  1.99  160,000  3.3086E−5  4.18 
DOF  E: SBP2  Rate  DOF  E: SBP4  Rate 

10,537  3.2863E−1  –  10,537  2.8321E−1  – 
40,905  1.1075E−1  1.57  40,905  3.9277E−2  2.85 
161,161  3.5769E−2  1.63  161,161  3.7081E−3  3.40 
Convergence in the maximum norm (60), for the second and fourthorder versions of each method, applied to Test Problem 2C (TP–2C), with diffusion coefficients \((\lambda _1, \lambda _2) = (1000, 1)\), and timestep \(\Delta t = 0.5h\). (DPM2I/DPM4I refers to the extension of the DPM method, to consider implicit geometry.)
DOF  E: CUT2  Rate  DOF  E: CUT4  Rate 

9988  6.4110E−1  –  10129  7.1811E−2  – 
39988  1.6506E−1  1.92  39952  3.9995E−3  4.08 
159988  3.8719E−2  2.07  160729  2.8978E−4  3.71 
DOF  E: DPM2  Rate  DOF  E: DPM4  Rate 

10,000  1.1178E−1  –  10,000  1.1392E−3  – 
40,000  1.8941E−2  2.56  40,000  5.9291E−5  4.26 
160,000  4.0950E−3  2.21  160,000  3.2716E−6  4.18 
DOF  E: DPM2I  Rate  DOF  E: DPM4I  Rate 

10,000  1.0377E−1  –  10,000  1.0905E−3  – 
40,000  1.7727E−2  2.55  40,000  5.5494E−5  4.30 
160,000  3.8853E−3  2.19  160,000  3.0003E−6  4.21 
DOF  E: SBP2  Rate  DOF  E: SBP4  Rate 

10,537  1.0025E−1  –  10,537  5.9131E−3  – 
40,905  2.5318E−2  1.99  40,905  4.8624E−4  3.60 
161,161  6.3459E−3  2.00  161,161  3.5001E−5  3.80 
The plots of spatial error at the final time \(T = 1.0\), shown in Fig. 8, are representative of other tests (not included in this text) on a single circular domain. The error in the cutFEM solution presents largely at the boundary; the error in the DPM solution typically has smooth error, even for grid points very near \(\varGamma \); while the error in the SBP–SAT–FD solution is not smooth at interfaces introduced by the domain partitioning.
Regarding the maxnorm error in presented in Table 3 and Fig. 10, we see that the DPM has smaller maxnorm by more than an order of magnitude. We also observe that the convergence rate of the fourthorder SBP–SAT–FD is only three. This suboptimal convergence is inline with the error plot in Fig. 9c, which shows that the error at the corners of the domain is significantly larger than elsewhere. In addition, the error is only nonsmooth along the interfaces on the two diagonal lines of the domain. We have also measured the \(L_2\) error at the final time \(T=1.0\) (not reported in this work), and fourthorder convergence is obtained.
In Table 5 and Fig. 12, we see the numerical results for (TP–2C), which shows that our numerical methods are robust to large jumps in diffusion coefficients, the analytical solution, and/or the flux of the true solution. Also, observe that the errors from DPM2/DPM4 (explicit geometry) and DPM2I/DPM4I (implicit geometry) in Tables 3, 4 and 5 are almost identical, which demonstrates the robustness and flexibility of the DPM.
6 Discussion
There are many possible methods (Sect. 1) for the numerical approximation of PDE posed on irregular domains, or on composite domains with interfaces. In this work, we consider three such methods, designed for the highorder accurate numerical approximation of parabolic PDEs (1, 2 or 3–9). Each implementation was written, tested, and optimized by the authors most experienced with the method—the cutFinite Element Method (cutFEM) by G. Ludvigsson, S. Sticko, G. Kreiss; the Difference Potentials Method (DPM) by K. R. Steffen, Q. Xia, Y. Epshteyn; and the Finite Difference Method satisfying SummationByParts, with a Simultaneous Approximation Term (SBP–SAT–FD) by S. Wang, G. Kreiss. Although we consider only one type of boundary/interface (a circle), we hope that the benchmark problems considered will be a valuable resource, and the numerical results a valuable comparison, for researchers interested in numerical methods for such problems.
The primary differences between the cutFEM and the standard finite element method are the stabilization terms for nearboundary degrees of freedom, and the quadrature over cut (intersected) elements. Tuning the free parameters in the stabilization terms could mitigate the errors observed in Figs. 8, 9. (We have done some preliminary experiments suggesting that the errors decrease when tuning these parameters, but further investigations are required in order to guarantee robustness.) Given a levelset description of the geometry, there are robust algorithms for constructing the quadrature over cut elements. Together, these differences allow for an immersed (nonconforming) grid to be used. The theoretical base for cutFEM is well established.
The DPM is based on the equivalence between the discrete system of Eq. (33) and the Boundary Equations with Projection (Theorem 1). The formulation outlined in Sect. 3.2 allows for an immersed (nonconforming) grid; fast \(\mathcal {O}(N \log N)\) algorithms, even for problems with general, smooth geometry; and reduces the size of the system to be solved at each timestep. The convergence theory is wellestablished for general, linear, elliptic boundary value problems, and we conjecture in Sect. 3.2 that this extends to the current setting. In this work, we have extended DPM to work with implicitlydefined geometries for the first time. This is a first step for solving problems where the interface moves with time.
In the finite difference framework (the SBP–SAT–FD method, in this work), the SBP property makes it possible to prove stability and convergence for highorder methods by an energy method. Combined with the SAT method to impose boundary and interface conditions, the SBP–SAT–FD method can be efficient to solve timedependent PDE. Geometrical features are resolved by curvilinear mapping, which requires an explicit parameterization of boundaries and interfaces. High quality grid generation is important—our experiments, though not reported in this work, have shown that the error in the solution is sensitive to both the orthogonality of the grid and the grid stretching.
Similarities between the cutFEM and the DPM (beyond the use of an immersed grid) include the thin layer of cut cells along the boundaries/interfaces (cutFEM) and the discrete grid boundary \(\gamma \) (DPM); and the use of higherorder normal derivatives in the stabilization term (cutFEM) and extension operator (in the Boundary Equations with Projection; DPM). A similarity between the cutFEM and SBP–SAT–FD is the weak imposition of boundary conditions, via Nitsche’s method (cutFEM) or the SAT method (SBP–SAT–FD). In this work, the DPM and the SBP–SAT–FD method both use an underlying finitedifference discretization, but the DPM is not restricted to this type of discretization.
Although both the cutFEM and the DPM use higherorder normal derivatives in their treatment of the boundary/interface, the precise usage differs. For cutFEM, it is the normal of the element interfaces cut by \(\varGamma \), while for DPM, it is the normal of the boundary/interface \(\varGamma \). Moreover, in the cutFEM, stabilization terms (19) involving higherorder normal derivatives at the boundaries of cutelements are added to the weak form of the PDE, to control the condition number of the mass and stiffness matrices, with a priori estimation of parameters to guarantee positivedefiniteness of these matrices; while in the DPM, the Boundary Equations with Projection is combined with the Extension Operator (Definition 4), which incorporates higherorder normal derivatives at the boundary/interface \(\varGamma \).
Returning to Sect. 5.3, we see (in Tables 1, 2, 3, 4, 5 and Figs. 6, 7, 8, 9, 10, 11, 12) that the expected rate of convergence for the second and fourthorder versions of DPM and cutFEM is achieved, while the DPM has the smallest error constant across all tests. For the SBP–SAT–FD method, expected convergence rates are obtained in some experiments. A noticeable exception is Test Problem 2A, for which the fourthorder SBP–SAT–FD method only has a convergence rate of three.
From the error plot in Fig. 9c, we observe that the large error is localized at the four corners of the domain \(\varOmega \), where the curvilinear grid is nonorthogonal and is stretched the most (see Fig. 5b).
As seen in the error plots (Figs. 8, 9), the error for the cutFEM and the SBP–SAT–FD has “spikes”, while for the DPM the error is smooth. A surprising observation from Fig. 9 is that conforming grids (on which the SBP–SAT–FD method is designed) do not necessarily produce more accurate solutions than immersed grids (on which the cutFEM and the DPM are designed). Indeed, it is challenging to construct a highquality curvilinear grid for the considered composite domain problem.
Future directions we hope to consider (in the context of new developments and also further comparisons) include: (i) parabolic problems with moving boundaries/interfaces, (ii) comparison of numerical methods for interface problems involving wave equations [12, 70, 71, 75, 78], (iii) extending our methods to consider PDEs in 3D, (iv) design of fast algorithms, and (v) design of adaptive versions of our methods.
Indeed, for (i), difficulties for the cutFEM might be the costly construction of quadrature, while for DPM difficulties might be the accurate construction of extension operators. Regarding (iii), this has already been done for the cutFEM and SBP–SAT–FD; while for the DPM, this is current work, with the main steps extending from 2D to 3D in a straightforward manner.
7 Conclusion
In this work, we propose a set of benchmark problems to test numerical methods for parabolic partial differential equations in irregular or composite domains, in the simplified geometric setting of Sect. 2, with the interface defined either explicitly or implicitly. Next, we compare and contrast three methods for the numerical approximation of such problems: the (i) cutFEM; (ii) DPM; and (iii) SBP–SAT–FD. Brief introductions of the three numerical methods are given in Sect. 3. It is noteworthy that the DPM has, for the first time, been extended to problems with an implicitlydefined interface.
For the three methods, the numerical results in Sect. 5.3 illustrate the highorder accuracy. Similar errors (different by a constant factor) are observed at grid points away from the boundary/interface, while the observed errors near the boundary/interface vary depending upon the given method. Although we consider only test problems with circular boundary/interface, the ideas underlying the three methods can readily be extended to more general geometries.
In general, all three methods require an accurate and efficient resolution of the explicitly or implicitlydefined irregular geometry: cutFEM relies on accurate quadrature rules for cut elements, and a good choice of stabilization parameters; DPM relies on an accurate and efficient representation of Cauchy data using a good choice of basis functions; and SBP–SAT–FD relies on the smooth parametrization to generate a highquality curvilinear grid.
Notes
Acknowledgements
The authors are grateful to the anonymous referees for their valuable remarks and questions, which led to significant improvements to the manuscript. We gratefully acknowledge the support of the Swedish Research Council (Grant No. 20146088); the Swedish Foundation for International Cooperation in Research and Higher Education (Grant No. STINTIB20166512); Uppsala University, Department of Information Technology; and the University of Utah, Department of Mathematics. Y. Epshteyn, K. R. Steffen, and Q. Xia also acknowledge partial support of Simons Foundation Grant No. 415673.
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