PDE Methods Research Articles

Convergence of an adaptive C0-interior penalty Galerkin method for the biharmonic problem

Abstract We develop a basic convergence analysis for an adaptive ${C}^{0}\mathsf{IPG}$ method for the biharmonic problem that provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the previous convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, here we have to deal with the fact that the Lagrange finite element spaces may possibly contain no proper $C^{1}$-conforming subspace. This prevents from a straight forward generalization and requires the development of some new key technical tools.

Error analysis of kernel/GP methods for nonlinear and parametric PDEs

We introduce a priori Sobolev-space error estimates for the solution of arbitrary nonlinear, and possibly parametric, PDEs that are defined in the strong sense, using Gaussian process and kernel based methods. The primary assumptions are: (1) a continuous embedding of the reproducing kernel Hilbert space of the kernel into a Sobolev space of sufficient regularity; and (2) the stability of the differential operator and the solution map of the PDE between corresponding Sobolev spaces. The proof is articulated around Sobolev norm error estimates for kernel interpolants and relies on the minimizing norm property of the solution. The error estimates demonstrate dimension-benign convergence rates if the solution space of the PDE is smooth enough. We illustrate these points with applications to high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some recent machine learning methods have been presented as breaking the curse of dimensionality in solving high-dimensional PDEs, our analysis suggests a more nuanced picture: there is a trade-off between the regularity of the solution and the presence of the curse of dimensionality. Therefore, our results are in line with the understanding that the curse is absent when the solution is regular enough.

Variational methods for some singular stochastic elliptic PDEs

Variational methods for some singular stochastic elliptic PDEs

Combining approach of collocation and finite difference methods for fractional parabolic PDEs

This research aims to estimate the solutions of fractional-order partial differential equations of spacial fractional and both time-space fractional order. For this, we use finite differences for time derivatives and the well-known collocation method for space derivatives with lower-order Bernstein polynomials as basis functions. We explain the mathematical formulations in detail. Convergence and stability analysis of the space–time fractional diffusion equation with the source term is reported subsequently. Three numerical examples are considered for demonstrating the accuracy and reliability of the proposed method.

SMS: Spiking marching scheme for efficient long time integration of differential equations

We propose a Spiking Neural Network (SNN)-based explicit numerical scheme for long time integration of time-dependent Ordinary and Partial Differential Equations (ODEs, PDEs). The core element of the method is a SNN, trained to use spike-encoded information about the solution at previous timesteps to predict spike-encoded information at the next timestep. After the network has been trained, it operates as an explicit numerical scheme that can be used to compute the solution at future timesteps, given a spike-encoded initial condition. A decoder is used to transform the evolved spiking-encoded solution back to function values. We present results from numerical experiments of using the proposed method for ODEs and PDEs of varying complexity.

Optimal error estimates of a non-uniform IMEX-L1 finite element method for time fractional PDEs and PIDEs

Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint elliptic part having (space-time) variable coefficients. The proposed scheme is based on a combination of an IMEX-L1 method on graded mesh in the temporal direction and a finite element method in the spatial direction. With the help of a discrete fractional Grönwall inequality, global almost optimal error estimates in L2- and H1-norms are derived for the problem with initial data u0∈H01(Ω)∩H2(Ω). The novelty of our approach is based on managing the interaction of the L1 approximation of the fractional derivative and the time discrete elliptic operator to derive the optimal estimate in H1-norm directly. Furthermore, a super convergence result is established when the elliptic operator is self-adjoint with time and space varying coefficients, and as a consequence, an L∞ error estimate is obtained for 2D problems that too with the initial condition is in H01(Ω)∩H2(Ω). All results proved in this paper are valid uniformly as α→1−, where α is the order of the Caputo fractional derivative. Numerical experiments are presented to validate our theoretical findings.

A robust collocation method for time fractional PDEs based on mean value theorem and cubic B-splines

This paper explains and applies a numerical technique utilizing the cubic B-spline functions and the mean value theorem (MVT) to solve a general time fractional partial differential equation (FPDE). The MVT for integrals enables us to approximate the time fractional derivatives in an appropriate simple form. We use the cubic B-spline functions to construct the numerical solution and its spatial derivatives. The great advantage of our technique is that it enables us to approximate solutions of many time FPDEs for several choices of F(x, t, u, ux, uxx). Two numerical examples have been included to emphasize the accuracy and efficiency of the method. It is demonstrated that the numerical method is unconditionally stable by employing the Von Neumann method (VNM).

A novel and simple spectral method for nonlocal PDEs with the fractional Laplacian

We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian (−Δ)α2. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol |ξ|α as the fractional Laplacian (−Δ)α2 at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This unique feature distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature which are usually limited to periodic boundary conditions (see Remark 1.1). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector multiplications. The computational complexity is O(2Nlog⁡(2N)), and the memory storage is O(N) with N the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems.

Review on Jacobi-Galerkin Spectral Method for Linear PDEs in Applied Mathematics

This study explores the spectral Galerkin approach to solving the space-time Schrödinger, wave, Airy, and beam equations. In order to facilitate the creation of a semi-analytical approximation solution, it uses polynomial bases that are formed from a linear combination of Jacobi polynomials (JPs) in both spatial and temporal dimensions. By using these polynomials to expand the exact solution, the paper hopes to satisfy the homogeneous starting and Dirichlet boundary requirements. Notably, the Jacobi Galerkin (JG) method exhibits exponential convergence rates if the solution is sufficiently smooth. This result emphasizes the JG approach' s potential as an effective numerical solution method, which has promise for a variety of applications in other domains where these equations occur, such as quantum mechanics, acoustics, optics, and structural mechanics.

Novel high-order explicit energy-preserving schemes for NLS-type equations based on the Lie-group method

In this paper, by taking the NLS-type equations as examples, we propose a novel high-order explicit energy-preserving Lie-group method for Hamiltonian PDEs. The main idea is to combine recently developed generalized SAV (G-SAV) approach (Cheng et al., 2021) and explicit Lie group method, in which the fourth-order or higher-order Lie group method only composes two exponentials in each stage, and this is the key point that the proposed methods can preserve energy of Hamiltonian PDEs. This is our first attempt to construct high-order explicit energy-preserving methods without using projection technique (Hairer et al., 2006 [14]), and some stability conclusions of our proposed methods for the NLS-type equations are also presented. Finally, we present 2D and 3D numerical simulations to demonstrate the stability and accuracy.

Decomposing Method for Space-Time Fractional Order PDEs

Numerical solutions to integer-order differential equations frequently employ decomposition as one of many methods. This paper provides a generalization of integer-order differential equations into fractional order, which is more general, as well as generalizing the decomposing method into the fractional case and then applying it to solve differential equations of fractional order in both space and time. A fractional calculus's theorems and properties to generalize both differential equations and the decomposition method have been utilized. This method to solve different fractional time and space partial differential equations, demonstrating its ability and applicability to various problems have been applied. The method to various cases to highlight its strengths has been used. The graphs and tables of results obtained using Matlab programs have been presented and discussed.

Reduced Variance Random Batch Methods for Nonlocal PDEs

Random Batch Methods (RBM) for mean-field interacting particle systems enable the reduction of the quadratic computational cost associated with particle interactions to a near-linear cost. The essence of these algorithms lies in the random partitioning of the particle ensemble into smaller batches at each time step. The interaction of each particle within these batches is then evolved until the subsequent time step. This approach effectively decreases the computational cost by an order of magnitude while increasing the amount of fluctuations due to the random partitioning. In this work, we propose a variance reduction technique for RBM applied to nonlocal PDEs of Fokker-Planck type based on a control variate strategy. The core idea is to construct a surrogate model that can be computed on the full set of particles at a linear cost while maintaining enough correlations with the original particle dynamics. Examples from models of collective behavior in opinion spreading and swarming dynamics demonstrate the great potential of the present approach.

An Efficient Replacement for Schrödinger's Partial Differential Equation

The classical finite difference method for solving time-dependent partial differential equations has become quite tedious and requires the use of off-the-shelf algorithms such as those in MATLAB. The treatment by finite difference method then solution of the n resulting first order algebraic equations is quite difficult since the underlying matrix is singular. We propose an alternative revolutionary statistical technique using Bmatrix chains, which are a product of the Cairo techniques. This technique is valid for both classical macroscopic physics such as the heat diffusion equation and modern microscopic quantum mechanics such as Schrödinger's PDE. He completely neglects partial differential equations as if they never existed. The numerical results of the proposed numerical statistical method show stability, accuracy and superiority over conventional PDE methods.

An optimization-based method for sign-changing elliptic PDEs

We study the numerical approximation of sign-shifting problems of elliptic type. We fully analyze and assess the method briefly introduced in [Abdulle, Huber, Lemaire; CRAS, 17]. Our method is based on domain decomposition and optimization. Upon an extra integrability assumption on the exact normal flux trace along the sign-changing interface, our method is proved to be convergent as soon as, for a given loading, the PDE admits a unique solution of finite energy. Departing from the T-coercivity approach, which relies on the use of geometrically fitted mesh families, our method works for arbitrary (interface-compliant) mesh sequences. Moreover, it is shown convergent for a class of problems for which T-coercivity is not applicable. A comprehensive set of test-cases complements our analysis.

Error assessment of an adaptive finite elements—neural networks method for an elliptic parametric PDE

We present a finite elements—neural network approach for the numerical approximation of parametric partial differential equations. The algorithm generates training data from finite element simulations, and uses a data-driven (supervised) feedforward neural network for the online approximation of the solution. The objective is to ensure that the overall error of the method is below some preset tolerance, and we thus control and balance the error coming from the finite element method, and the one introduced by the neural network approximation.Two finite element methods are considered and compared; a fixed grid approach uses the same mesh for all values of the parameters, while an adaptive finite elements approach enforces the same discretization error uniformly in the parameters space.Numerical results are presented for an elliptic model problem. The fixed grid approach shows limitations in terms of error balancing and control, while the adaptive approach allows a better accuracy and more flexibility of the method. We conclude by proposing an adaptive algorithm to control the size of the training set given a network architecture and ensure that the overall error of the method is below a given tolerance.

A Priori Error Estimate of Deep Mixed Residual Method for Elliptic PDEs

A Priori Error Estimate of Deep Mixed Residual Method for Elliptic PDEs

Numerical Methods for Fourth-Order PDEs on Overlapping Grids with Application to Kirchhoff–Love Plates

Numerical Methods for Fourth-Order PDEs on Overlapping Grids with Application to Kirchhoff–Love Plates

Randomized reduced basis methods for parameterized fractional elliptic PDEs

Randomized reduced basis methods for parameterized fractional elliptic PDEs

Decay rates of convergence for Fokker-Planck equations with confining drift

We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or sub-exponential decay rates, as time goes to infinity, of zero average solutions, under some diffusivity condition on the Levy process, which includes the fractional Laplace operator as a model example. Our approach relies on the long time oscillation estimates of the adjoint problem and applies to (the possible superposition of) both local and nonlocal diffusions, as well as to strongly or weakly confining drifts. Our results extend, with a unifying perspective, many previous works based on different analytic or probabilistic methods, with several interesting connections. On one hand, we make a link between the (nonlinear) PDE methods used for the long time behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations; on another hand, we give a purely analytical approach towards some oscillation decay estimates which were obtained so far only with probabilistic coupling methods.

A correction function-based kernel-free boundary integral method for elliptic PDEs with implicitly defined interfaces

This work addresses a novel version of the kernel-free boundary integral (KFBI) method for solving elliptic PDEs with implicitly defined irregular boundaries and interfaces. We focus on boundary value problems and interface problems, which are reformulated into boundary integral equations and solved with the matrix-free GMRES method. In the KFBI method, evaluating boundary and volume integrals only requires solving equivalent but much simpler interface problems in a bounding box, for which fast solvers such as FFTs and geometric multigrid methods are applicable. For the simple interface problem, a correction function is introduced for both the evaluation of right-hand side correction terms and the interpolation of a non-smooth potential function. A mesh-free collocation method is proposed to compute the correction function near the interface. The new method avoids complicated derivation for derivative jumps of the solution and is easy to implement, especially for the fourth-order method in three space dimensions. Various numerical examples are presented, including challenging cases such as high-contrast coefficients, arbitrarily close interfaces and heterogeneous interface problems. The reported numerical results verify that the proposed method is both accurate and efficient.