Showing papers in "SIAM Journal on Numerical Analysis in 2014"
TL;DR: In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed.
Abstract: In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis.
293 citations
TL;DR: A novel stream formulation of the virtual element method (VEM) for the solution of the Stokes problem is proposed and analyzed and it is equivalent to the velocity-pressure (inf-sup stable) mimetic scheme presented.
Abstract: In this paper we propose and analyze a novel stream formulation of the virtual element method (VEM) for the solution of the Stokes problem. The new formulation hinges upon the introduction of a suitable stream function space (characterizing the divergence free subspace of discrete velocities) and it is equivalent to the velocity-pressure (inf-sup stable) mimetic scheme presented in [L. Beirao da Veiga et al., J. Comput. Phys., 228 (2009), pp. 7215--7232] (up to a suitable reformulation into the VEM framework). Both schemes are thus stable and linearly convergent but the new method results to be more desirable as it employs much less degrees of freedom and it is based on a positive definite algebraic problem. Several numerical experiments assess the convergence properties of the new method and show its computational advantages with respect to the mimetic one.
238 citations
TL;DR: An abstract mathematical framework is established rigorously here together with applications to the numerical solution of both nonlocal models and their local limits and a general state-based peridynamic system parametrized by the horizon radius.
Abstract: Many problems in nature, being characterized by a parameter, are of interest both with a fixed parameter value and with the parameter approaching an asymptotic limit. Numerical schemes that are convergent in both regimes offer robust discretizations, which can be highly desirable in practice. The asymptotically compatible schemes studied in this paper meet such objectives for a class of parametrized problems. An abstract mathematical framework is established rigorously here together with applications to the numerical solution of both nonlocal models and their local limits. In particular, the framework can be applied to nonlocal diffusion models and a general state-based peridynamic system parametrized by the horizon radius. Recent findings have exposed the risks associated with some discretizations of nonlocal models when the horizon radius is proportional to the discretization parameter. Thus, it is desirable to develop asymptotically compatible schemes for such models so as to offer robust numerical dis...
165 citations
TL;DR: The core object of this paper is to derive a class of fourth order approximations, called the weighted and shifted Lubich difference operators, for space fractional derivatives, and use the derived schemes to improve the accuracy of higher order schemes.
Abstract: Because of the nonlocal properties of fractional operators, higher order schemes play a more important role in discretizing fractional derivatives than classical ones. The striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first order schemes but greatly improve the accuracy. Nowadays, there are already two types of second order discretization schemes for space fractional derivatives: the first type is given and discussed in [Sousa and Li, arXiv:1109.2345v1, 2011; Chen and Deng, Appl. Math. Model., 38 (2014), pp. 3244--3259; Chen, Deng, and Wu, Appl. Numer. Math., 70 (2013), pp. 22--41]; and the second type is a class of schemes presented in [Tian, Zhou, and Deng, Math. Comp., to appear; also available online from arXiv:1201.5949, 2012]. The core object of this paper is to derive a class of fourth order approximations, called the weighted and shifted Lubich difference operators, for space fractional derivatives. Then we use the derived schemes t...
150 citations
TL;DR: In this article, a numerical method for the fractional Laplacian was proposed, based on the singular integral representation for the operator, which combines finite differences with numerical quadrature.
Abstract: The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a nonlocal operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite differences with numerical quadrature to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solutions with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.
143 citations
TL;DR: It is proved stability and optimal order of convergence ${\cal O}(h^{k+1})$ for the fractional diffusion problem, and an orders of convergence of h^{k +\frac{1}{2}})$ is established for the general fractional convection-diffusion problem.
Abstract: We propose a discontinuous Galerkin method for fractional convection-diffusion equations with a superdiffusion operator of order $\alpha (1<\alpha<2)$ defined through the fractional Laplacian. The fractional operator of order $\alpha$ is expressed as a composite of first order derivatives and a fractional integral of order $2-\alpha$. The fractional convection-diffusion problem is expressed as a system of low order differential/integral equations, and a local discontinuous Galerkin method scheme is proposed for the equations. We prove stability and optimal order of convergence ${\cal O}(h^{k+1})$ for the fractional diffusion problem, and an order of convergence of ${\cal O}(h^{k+\frac{1}{2}})$ is established for the general fractional convection-diffusion problem. The analysis is confirmed by numerical examples.
126 citations
TL;DR: A time-stepping discontinuous Petrov--Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems is proposed and analyzed and the existence, uniqueness, and stability of approximate solutions are proved.
Abstract: We propose and analyze a time-stepping discontinuous Petrov--Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence, uniqueness, and stability of approximate solutions and derive error estimates. To achieve high order convergence rates from the time discretizations, the time mesh is graded appropriately near $t=0$ to compensate for the singular (temporal) behavior of the exact solution near $t=0$ caused by the weakly singular kernel, but the spatial mesh is quasi uniform. In the $L_\infty((0,T);L_2(\Omega))$-norm, ($(0,T)$ is the time domain and $\Omega$ is the spatial domain); for sufficiently graded time meshes, a global convergence of order $k^{m+\alpha/2}+h^{r+1}$ is shown, where $0<\alpha<1$ is the fractional exponent, $k$ is the maximum time step, $h$ is the maximum diameter of the elements of the spatial mesh, and $m$ and $r$ are the degrees of approximate solutions in time an...
111 citations
TL;DR: An unconditionally optimal error analysis of an uncoupled and linearized Crank--Nicolson Galerkin finite element method for the time-dependent nonlinear thermistor equations in d-dimensional space is presented.
Abstract: This paper focuses on unconditionally optimal error analysis of an uncoupled and linearized Crank--Nicolson Galerkin finite element method for the time-dependent nonlinear thermistor equations in $d$-dimensional space, $d=2,3$. In our analysis, we split the error function into two parts, one from the spatial discretization and one from the temporal discretization, by introducing a corresponding time-discrete (elliptic) system. We present a rigorous analysis for the regularity of the solution of the time-discrete system and error estimates of the time discretization. With these estimates and the proved regularity, optimal error estimates of the fully discrete Crank--Nicolson Galerkin method are obtained unconditionally. Numerical results confirm our analysis and show the efficiency of the method.
101 citations
TL;DR: A modification of the fast-marching algorithm, which solves the anisotropic eikonal equation associated to an arbitrary continuous Riemannian metric on a two- or three-dimensional domain, and proves the convergence of the algorithm and illustrates its efficiency by numerical experiments.
Abstract: We introduce a modification of the Fast Marching Algorithm, which solves the generalized eikonal equation associated to an arbitrary continuous riemannian metric, on a two or three dimensional box domain. The algorithm has a logarithmic complexity in the maximum anisotropy ratio of the riemannian metric, which allows to handle extreme anisotropies for a reduced numerical cost. We establish that the output of the algorithm converges towards the viscosity solution of continuous problem, as the discretization step tends to zero. The algorithm is based on the computation at each grid point of a reduced basis of the unit lattice, with respect to the symmetric positive definite matrix encoding the desired anisotropy at this point.
101 citations
TL;DR: The method is proved to be consistent and stable, with convergence rates that are optimal with respect to mesh size, and suboptimal in the polynomial degree by only half an order.
Abstract: We propose an $hp$-version discontinuous Galerkin finite element method for fully nonlinear second-order elliptic Hamilton--Jacobi--Bellman equations with Cordes coefficients. The method is proved to be consistent and stable, with convergence rates that are optimal with respect to mesh size, and suboptimal in the polynomial degree by only half an order. Numerical experiments on problems with nonsmooth solutions and strongly anisotropic diffusion coefficients illustrate the accuracy and computational efficiency of the scheme. An existence and uniqueness result for strong solutions of the fully nonlinear problem and a semismoothness result for the nonlinear operator are also provided.
100 citations
TL;DR: As expected from the upper bound derivation, the $H^1$-norm of the error in the method converges at a linear rate with respect to the size of the mesh elements.
Abstract: We derive upper and lower bounds on the gradients of Wachspress coordinates defined over any simple convex $d$-dimensional polytope $P$. The bounds are in terms of a single geometric quantity $h_\ast$, which denotes the minimum distance between a vertex of $P$ and any hyperplane containing a nonincident face. We prove that the upper bound is sharp for $d=2$ and analyze the bounds in the special cases of hypercubes and simplices. Additionally, we provide an implementation of the Wachspress coordinates on convex polyhedra using MATLAB and employ them in a three-dimensional finite element solution of the Poisson equation on a nontrivial polyhedral mesh. As expected from the upper bound derivation, the $H^1$-norm of the error in the method converges at a linear rate with respect to the size of the mesh elements.
TL;DR: A systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic SDEs (Brownian and Langevin dynamics) with an accuracy independent of the weak order of the underlying method.
Abstract: We introduce new sufficient conditions for a numerical method to approximate with high order of accuracy the invariant measure of an ergodic system of stochastic differential equations, independently of the weak order of accuracy of the method. We then present a systematic procedure based on the framework of modified differential equations for the construction of stochastic integrators that capture the invariant measure of a wide class of ergodic SDEs (Brownian and Langevin dynamics) with an accuracy independent of the weak order of the underlying method. Numerical experiments confirm our theoretical findings.
TL;DR: This work considers an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann--Liouville type and order $\alpha\in (1,2)$.
Abstract: We consider an initial boundary value problem for a one-dimensional fractional-order parabolic equation with a space fractional derivative of Riemann--Liouville type and order $\alpha\in (1,2)$. We study a spatial semidiscrete scheme using the standard Galerkin finite element method with piecewise linear finite elements, as well as fully discrete schemes based on the backward Euler method and the Crank--Nicolson method. Error estimates in the $L^2(D)$- and $H^{\alpha/2}(D)$-norm are derived for the semidiscrete scheme and in the $L^2(D)$-norm for the fully discrete schemes. These estimates cover both smooth and nonsmooth initial data and are expressed directly in terms of the smoothness of the initial data. Extensive numerical results are presented to illustrate the theoretical results.
TL;DR: A general a posteriori error analysis is established for the natural norms of the DPG schemes under conditions equivalent to a priori stability estimates, proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors.
Abstract: A combination of ideas in least-squares finite element methods with those of hybridized methods recently led to discontinuous Petrov--Galerkin (DPG) finite element methods. They minimize a residual inherited from a piecewise ultraweak formulation in a nonstandard, locally computable, dual norm. This paper establishes a general a posteriori error analysis for the natural norms of the DPG schemes under conditions equivalent to a priori stability estimates. It is proven that the locally computable residual norm of any discrete function is a lower and an upper error bound up to explicit data approximation errors. The presented abstract framework for a posteriori error analysis applies to known DPG discretizations of Laplace and Lame equations and to a novel DPG method for the stress-velocity formulation of Stokes flow with symmetric stress approximations. Since the error control does not rely on the discrete equations, it applies to inexactly computed or otherwise perturbed solutions within the discrete space...
TL;DR: This work analyzes adaptive mesh-refining algorithms for conforming finite element discretizations of certain nonlinear second-order partial differential equations and proves convergence even with optimal algebraic convergence rates.
Abstract: We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain nonlinear second-order partial differential equations. We allow continuous polynomials of arbitrary but fixed polynomial order. The adaptivity is driven by the residual error estimator. We prove convergence even with optimal algebraic convergence rates. In particular, our analysis covers general linear second-order elliptic operators. Unlike prior works for linear nonsymmetric operators, our analysis avoids the interior node property for the refinement, and the differential operator has to satisfy a G\rarding inequality only. If the differential operator is uniformly elliptic, no additional assumption on the initial mesh is posed.
TL;DR: The derivations of existing error bounds for reduced order models of time varying partial differential equations (PDEs) constructed using proper orthogonal decomposition (POD) have relied on bounding the error between the POD data and various POD projections of that data, so this work considers time varying data taking values in two different Hilbert spaces.
Abstract: The derivations of existing error bounds for reduced order models of time varying partial differential equations (PDEs) constructed using proper orthogonal decomposition (POD) have relied on bounding the error between the POD data and various POD projections of that data. Furthermore, the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the POD data approximation error bounds. We consider time varying data taking values in two different Hilbert spaces $ H $ and $ V $, with $ V \subset H $, and prove exact expressions for the POD data approximation errors considering four different POD projections and the two different Hilbert space error norms. Furthermore, the exact error expressions can be computed using only the POD eigenvalues and modes, and we prove the errors converge to zero as the number of POD modes increases. We consider the POD error estimation approaches of Kunisch and Volkwein [SIAM J. Numer. Anal., 40 (2002), pp. 492--515] and Chapelle, Gariah, an...
TL;DR: An indirect finite element method is developed for the Dirichlet boundary-value problems of Caputo fractional differential equations, which reduces the computational work for the numerical solution of variable-coefficient fractional diffusion equations from N to O and the memory requirement on any quasiuniform space partition is reduced.
Abstract: We prove the wellposedness of the Galerkin weak formulation and Petrov--Galerkin weak formulation for inhomogeneous Dirichlet boundary-value problems of constant- or variable-coefficient conservative Caputo space-fractional diffusion equations. We also show that the weak solutions to their Riemann--Liouville analogues do not exist, in general. In addition, we develop an indirect finite element method for the Dirichlet boundary-value problems of Caputo fractional differential equations, which reduces the computational work for the numerical solution of variable-coefficient fractional diffusion equations from $O(N^3)$ to $O(N)$ and the memory requirement from $O(N^2)$ to $O(N)$ on any quasiuniform space partition. We further prove a nearly sharp error estimate for the method, which is expressed in terms of the smoothness of the prescribed data of the problem only. We carry out numerical experiments to investigate the performance of the method in comparison with the Galerkin finite element method.
TL;DR: It is proved that the derivative approximation of the DG solution is superconvergent with a rate $k+1$ at all interior left Radau points and for the domain average under quasi-uniform meshes and some suitable initial discretization.
Abstract: In this paper, we study superconvergence properties of the discontinuous Galerkin (DG) method for one-dimensional linear hyperbolic equations when upwind fluxes are used. We prove, for any polynomial degree $k$, the $2k+1$th (or $2k+1/2$th) superconvergence rate of the DG approximation at the downwind points and for the domain average under quasi-uniform meshes and some suitable initial discretization. Moreover, we prove that the derivative approximation of the DG solution is superconvergent with a rate $k+1$ at all interior left Radau points. All theoretical findings are confirmed by numerical experiments.
TL;DR: In this article, a fully discrete semi-Lagrangian scheme for a first order mean field game system is proposed and the resulting discretization admits at least one solution.
Abstract: In this work we propose a fully discrete semi-Lagrangian scheme for a first order mean field game system. We prove that the resulting discretization admits at least one solution and, in the scalar case, we prove a convergence result for the scheme. Numerical simulations and examples are also discussed.
TL;DR: The distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals are emphasized and it is shown how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension.
Abstract: Let $f(z)$ be an analytic or meromorphic function in the closed unit disk sampled at the $n$th roots of unity. Based on these data, how can we approximately evaluate $f(z)$ or $f^{(m)}(z)$ at a point $z$ in the disk? How can we calculate the zeros or poles of $f$ in the disk? These questions exhibit in the purest form certain algorithmic issues that arise across computational science in areas including integral equations, partial differential equations, and large-scale linear algebra. We analyze some of the possibilities and emphasize the distinction between algorithms based on polynomial or rational interpolation and those based on trapezoidal rule approximations of Cauchy integrals. We then show how these developments apply to the problem of computing the eigenvalues in the disk of a matrix of large dimension. Finally we highlight the power of rational in comparison with polynomial approximations for some of these problems.
TL;DR: This paper proposes an explicit, (at least) second-order, maximum principle sat- isfying, Lagrange finite element method for solving nonlinear scalar conservation equations.
Abstract: This paper proposes an explicit, (at least) second-order, maximum principle sat- isfying, Lagrange finite element method for solving nonlinear scalar conservation equations. The technique is based on a new viscous bilinear form introduced in Guermond and Nazarov (Com- put. Methods Appl. Mech. Engrg., 272 (2014), pp. 198-213), a high-order entropy viscosity method, and the Boris-Book-Zalesak flux correction technique. The algorithm works for arbitrary meshes in any space dimension and for all Lipschitz fluxes. The formal second-order accuracy of the method and its convergence properties are tested on a series of linear and nonlinear benchmark problems.
TL;DR: The idea is to recast solutions as fixed points of an operator defined on a Banach space of rapidly decaying Chebyshev coefficients and to use the so-called radii polynomials to show the existence of a unique fixed point near an approximate solution.
Abstract: A computational method based on Chebyshev series to rigorously compute solutions of initial and boundary value problems of analytic nonlinear vector fields is proposed. The idea is to recast solutions as fixed points of an operator defined on a Banach space of rapidly decaying Chebyshev coefficients and to use the so-called radii polynomials to show the existence of a unique fixed point near an approximate solution. As applications, solutions of initial value problems in the Lorenz equations and symmetric connecting orbits in the Gray--Scott equation are rigorously computed. The symmetric connecting orbits are obtained by solving a boundary value problem with one of the boundary values in the stable manifold.
TL;DR: In this article, a quasi-Monte Carlo method was proposed to approximate the expected values of linear functionals of Petrov-Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters.
Abstract: We construct quasi--Monte Carlo methods to approximate the expected values of linear functionals of Petrov--Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in the numerical solution of differential and integral equations with random field inputs. We analyze the regularity of the solutions with respect to the parameters in terms of the rate of decay of the fluctuations of the input field. If $p\in (0,1]$ denotes the “summability exponent” corresponding to the fluctuations in affine-parametric families of operators, then we prove that deterministic “interlaced polynomial lattice rules” of order $\alpha = \lfloor 1/p \rfloor+1$ in $s$ dimensions with $N$ points can be constructed using a fast component-by-component algorithm, in $\mathcal{O}(\alpha\,s\, N\log N + \alpha^2\,s^2 N)$ operations, to achieve a convergence rate of $\mathcal{O}(N^{-1/p})$, with the implied constant independent of $s$. This dimension-independen...
TL;DR: It is shown that, under a sufficient condition on the coefficients, the schemes enjoy a fundamental stability property, and an analysis of the truncation error allows us to design approximation with arbitrary order of convergence.
Abstract: We study the convergence rate of a class of linear multistep methods for backward stochastic differential equations (BSDEs). We show that, under a sufficient condition on the coefficients, the schemes enjoy a fundamental stability property. Coupling this result to an analysis of the truncation error allows us to design approximation with arbitrary order of convergence. Contrary to the analysis performed in [W. Zhao, G. Zhang, and L. Ju, SIAM J. Numer. Anal., 48 (2010), pp. 1369--1394], we consider general diffusion models and BSDEs with driver depending on $z$. The class of methods we consider contains well-known methods from the ODE framework as Nystrom, Milne, or Adams methods. Finally, we provide a numerical illustration of the convergence of some methods.
TL;DR: A discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous Galerkin (DG) space-time finite element discretization.
Abstract: In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in $\Bbb{R}^d$ defines a $d$-dimensional space-time manifold in the space-time continuum $\Bbb{R}^{d+1}$. We derive and analyze a variational formulation for a class of diffusion problems on the space-time manifold. For this variational formulation new well-posedness and stability results are derived. The analysis is based on an inf-sup condition and involves some natural, but nonstandard, (anisotropic) function spaces. Based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous Galerkin (DG) space-time finite element discretization. This DG space-time method is explained and results of numerical experiments are presented that illustrate its properties.
TL;DR: In this paper, it was shown that as the size of the least square problem approaches infinity, the computed approximation converges to the true derivative, assuming that the solution is a constrained least square.
Abstract: For a parameterized hyperbolic system $u_{i+1} = f(u_i,s)$, the derivative of an ergodic average $\langle J\rangle = \lim_{n\rightarrow\infty} \frac1n \sum_1^n J(u_i,s)$ to the parameter $s$ can be computed via the least squares shadowing method. This method solves a constrained least squares problem and computes an approximation to the desired derivative $\frac{d\langle J\rangle}{ds}$ from the solution. This paper proves that as the size of the least squares problem approaches infinity, the computed approximation converges to the true derivative.
TL;DR: It is proved that the $L^2$ error bound the characteristic time-discrete system of the fully discrete method of characteristics to the time- Discrete system is $\tau$-independent and the numerical solution is bounded in $W^{1,\infty}$-norm unconditionally.
Abstract: The method of characteristics type is especially effective for convection-dominated diffusion problems. Due to the nature of characteristic temporal discretization, the method allows one to use a large time step in many practical computations, while all previous theoretical analyses always required certain restrictions on the time stepsize. Here, we present a new analysis to establish unconditionally optimal error estimates for a modified method of characteristics with a mixed finite element approximation to the miscible displacement problem in $\mathbb{R}^d\ (d=2,3)$. For this purpose, we introduce a new characteristic time-discrete system. We prove that the $L^2$ error bound the characteristic time-discrete systemof the fully discrete method of characteristics to the time-discrete system is $\tau$-independent and the numerical solution is bounded in $W^{1,\infty}$-norm unconditionally. With the boundedness, optimal error estimates are established in a traditional manner. Numerical results confirm our th...
TL;DR: A modified finite element method that is able to approximate interface problems with high accuracy is presented, and optimal order of convergence for elliptic problems is shown and a bound on the condition number of the system matrix is given.
Abstract: We present a modified finite element method that is able to approximate interface problems with high accuracy. We consider interface problems where the solution is continuous; its derivatives, however, may be discontinuous across interface curves within the domain. The proposed discretization is based on a local modification of the finite element basis functions using a fixed quadrilateral mesh. Instead of moving mesh nodes, we resolve the interface locally by an adapted parametric approach. All modifications are applied locally and in an implicit fashion. The scheme is easy to implement and is well suited for time-dependent moving interface problems. We show optimal order of convergence for elliptic problems, and further, we give a bound on the condition number of the system matrix. Both estimates do not depend on the interface location relative to the mesh.
TL;DR: This paper shows that the optimal order of accuracy in the WENO reconstruction is achieved for h^3 in the smooth case and $h^2 near discontinuities) for $\varepsilon = K h^q$ with $q \le 3$ and $pq \ge ...
Abstract: Recently, Arandiga et al. showed in [SIAM J. Numer. Anal., 49 (2011), pp. 893--915] for a class of weighted ENO (WENO) schemes that the parameter $\varepsilon$ occurring in the smoothness indicators of the scheme should be chosen proportional to the square of the mesh size, $h^2$, to achieve the optimal order of accuracy. Unfortunately, these results cannot be applied to the compact third order WENO reconstruction procedure introduced in [D. Levy, G. Puppo, and G. Russo, SIAM J. Sci. Comput., 22 (2000), pp. 656--672], which we apply within the semidiscrete central scheme of [A. Kurganov and D. Levy, SIAM J. Sci. Comput., 22 (2000), pp. 1461--1488], a commonly used scheme for the numerical solution of conservation laws and convection-diffusion equations. The aim of this paper is to close this gap. In particular, we will show that we achieve the optimal order of accuracy in the WENO reconstruction ($h^3$ in the smooth case and $h^2$ near discontinuities) for $\varepsilon = K h^q$ with $q \le 3$ and $pq \ge ...
TL;DR: An error analysis of an Eulerian finite element method for solving parabolic partial differential equations (PDEs) posed on evolving hypersurfaces is presented and regularity results for solutions of parabolic PDEs on an evolving surface are derived.
Abstract: In this paper we present an error analysis of an Eulerian finite element method for solving parabolic partial differential equations (PDEs) posed on evolving hypersurfaces in Rd, d =2 , 3. The method employs discontinuous piecewise linear in time-continuous piecewise linear in space finite elements and is based on a space-time weak formulation of a surface PDE problem. Trial and test surface finite element spaces consist of traces of standard volumetric elements on a space-time manifold resulting from the evolution of a surface. We prove first order convergence in space and time of the method in an energy norm and second order convergence in a weaker norm. Furthermore, we derive regularity results for solutions of parabolic PDEs on an evolving surface, which we need in a duality argument used in the proof of the second order convergence estimate.