Solving partial differential equations on (evolving) surfaces with radial basis functions

Wendland, Holger; Künemund, Jens

doi:10.1007/s10444-020-09803-0

Cited by 13 publications

(5 citation statements)

References 42 publications

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“…In more general evolutions of surfaces, the velocity is often coupled with the solution of the PDE on the surfaces, such as the model of solid tumor growth [ 3 , 4 , 16 , 17 ]. This model is mathematically described by the following reaction–diffusion system [ 16 ]:

where

…”

Section: Numerical Experimentsmentioning

confidence: 99%

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Numerical Study on an RBF-FD Tangent Plane Based Method for Convection–Diffusion Equations on Anisotropic Evolving Surfaces

Adil

Xiao

Feng

2022

Entropy

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In this paper, we present a fully Lagrangian method based on the radial basis function (RBF) finite difference (FD) method for solving convection–diffusion partial differential equations (PDEs) on evolving surfaces. Surface differential operators are discretized by the tangent plane approach using Gaussian RBFs augmented with two-dimensional (2D) polynomials. The main advantage of our method is the simplicity of calculating differentiation weights. Additionally, we couple the method with anisotropic RBFs (ARBFs) to obtain more accurate numerical solutions for the anisotropic growth of surfaces. In the ARBF interpolation, the Euclidean distance is replaced with a suitable metric that matches the anisotropic surface geometry. Therefore, it will lead to a good result on the aspects of stability and accuracy of the RBF-FD method for this type of problem. The performance of this method is shown for various convection–diffusion equations on evolving surfaces, which include the anisotropic growth of surfaces and growth coupled with the solutions of PDEs.

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where

…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…The RBF-FD methods have been successfully developed to convection–diffusion and reaction–diffusion PDEs on domains [ 10 , 11 ] and surfaces [ 12 , 13 , 14 , 15 ]. The RBF-FD methods have also been applied to the evolving surface PDEs [ 16 , 17 ].…”

Section: Introductionmentioning

confidence: 99%

Numerical Study on an RBF-FD Tangent Plane Based Method for Convection–Diffusion Equations on Anisotropic Evolving Surfaces

Adil

Xiao

Feng

2022

Entropy

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“…Many works have been devoted to the numerical solutions of semilinear elliptic problems such as finite element method (FEM) [2,3], finite difference method [4], finite volume element method [5] and discontinuous Galerkin method [6]. Recently, some collocation meshless (or meshfree) methods [7,8], Galerkintype meshless method [8] and generalized finite difference method [9,10] have been developed to solve the semilinear PDEs. Unlike mesh-based numerical methods, the shape functions used in the meshless methods [11][12][13][14] are linkage with nodes (or particles) scattered in the underlying computational domain, which reduces the dependence on the mesh.…”

Section: Introductionmentioning

confidence: 99%

A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

Li¹

2024

AAMM

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A Nitsche-based element-free Galerkin (EFG) method for solving semilinear elliptic problems is developed and analyzed in this paper. The existence and uniqueness of the weak solution for semilinear elliptic problems are proved based on a condition that the nonlinear term is an increasing Lipschitz continuous function of the unknown function. A simple iterative scheme is used to deal with the nonlinear integral term. We proved the existence, uniqueness and convergence of the weak solution sequence for continuous level of the simple iterative scheme. A commonly used assumption for approximate space, sometimes called inverse assumption, is proved. Optimal order error estimates in L 2 and H 1 norms are proved for the linear and semilinear elliptic problems. In the actual numerical calculation, the characteristic distance h does not appear explicitly in the parameter β introduced by the Nitsche method. The theoretical results are confirmed numerically.

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“…However, it may suffer from the intrinsic singularities that are built into the PDE formulation (e.g., poles in spherical coordinates) or in the boundary integral kernel involving Green's function; thus, it needs careful treatments near the singularities. A mesh-free approach called radial basis function method [12,13] works by first representing the solution by a linear combination of radial basis functions, and then substituting the approximation at some chosen points into the differential equation directly. As a result, a dense linear system of coefficients must be solved which is usually ill-conditioned.…”

Section: Introductionmentioning

confidence: 99%

A shallow physics-informed neural network for solving partial differential equations on surfaces

Hu¹,

Shih²,

Lin³

et al. 2022

Preprint

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In this paper, we introduce a mesh-free physics-informed neural network for solving partial differential equations on surfaces. Based on the idea of embedding techniques, we write the underlying surface differential equations using conventional Cartesian differential operators. With the aid of level set function, the surface geometrical quantities, such as the normal and mean curvature of the surface, can be computed directly and used in our surface differential expressions. So instead of imposing the normal extension constraints used in literature, we take the whole Cartesian differential expressions into account in our loss function. Meanwhile, we adopt a completely shallow (one hidden layer) network so the present model is easy to implement and train. We perform a series of numerical experiments on both stationary and time-dependent partial differential equations on complicated surface geometries. The result shows that, with just a few hundred trainable parameters, our network model is able to achieve high predictive accuracy.

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Solving partial differential equations on (evolving) surfaces with radial basis functions

Cited by 13 publications

References 42 publications

Numerical Study on an RBF-FD Tangent Plane Based Method for Convection–Diffusion Equations on Anisotropic Evolving Surfaces

Numerical Study on an RBF-FD Tangent Plane Based Method for Convection–Diffusion Equations on Anisotropic Evolving Surfaces

A Nitsche-Based Element-Free Galerkin Method for Semilinear Elliptic Problems

A shallow physics-informed neural network for solving partial differential equations on surfaces

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