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Register a new userOn explicit form of the FEM stiffness matrix for the integral fractional Laplacian on non-uniform meshes
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Added by
Changtao Sheng Phd
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We derive exact form of the piecewise-linear finite element stiffness matrix on general non-uniform meshes for the integral fractional Laplacian operator in one dimension, where the derivation is accomplished in the Fourier transformed space. With such an exact formulation at our disposal, we are able to numerically study some intrinsic properties of the fractional stiffness matrix on some commonly used non-uniform meshes (e.g., the graded mesh), in particular, to examine their seamless transition to those of the usual Laplacian.
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In this paper, we study diagonal dominance of the stiffness matrix resulted from the piecewise linear finite element discretisation of the integral fractional Laplacian under global homogeneous Dirichlet boundary condition in one spatial dimension. We first derive the exact form of this matrix in the frequency space which is extendable to multi-dimensional rectangular elements. Then we give the complete answer when the stiffness matrix can be strictly diagonally dominant. As one application, we apply this notion to the construction of maximum principle preserving schemes for the fractional-in-space Allen-Cahn equation, and provide ample numerical results to verify our findings.
In this paper, we provide a framework of designing the local discontinuous Galerkin scheme for integral fractional Laplacian $(-Delta)^{s}$ with $sin(0,1)$ in two dimensions. We theoretically prove and numerically verify the numerical stability and convergence of the scheme with the convergence rate no worse than $mathcal{O}(h^{k+frac{1}{2}})$.
We explore the connection between fractional order partial differential equations in two or more spatial dimensions with boundary integral operators to develop techniques that enable one to efficiently tackle the integral fractional Laplacian. In particular, we develop techniques for the treatment of the dense stiffness matrix including the computation of the entries, the efficient assembly and storage of a sparse approximation and the efficient solution of the resulting equations. The main idea consists of generalising proven techniques for the treatment of boundary integral equations to general fractional orders. Importantly, the approximation does not make any strong assumptions on the shape of the underlying domain and does not rely on any special structure of the matrix that could be exploited by fast transforms. We demonstrate the flexibility and performance of this approach in a couple of two-dimensional numerical examples.
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In this paper, we introduce two new families of generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate generalised Hermite spectral algorithms for PDEs with integral fractional Laplacian (IFL) and/or Schr{o}dinger operators in $mathbb R^d.$ As a generalisation of the G. Szeg{o}s family in 1D (1939), the first family of GHPs (resp. GHFs) are orthogonal with respect to $|bx|^{2mu} e^{-|bx|^2}$ (resp. $|bx |^{2mu}$) in $mathbb R^d$. We further define adjoint generalised Hermite functions (A-GHFs) which have an interwoven connection with the corresponding GHFs through the Fourier transform, and which are orthogonal with respect to the inner product $[u,v]_{H^s(mathbb R^d)}=((-Delta)^{s/ 2}u, (-Delta)^{s/2} v )_{mathbb R^d}$ associated with the IFL of order $s>0$. Thus, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds efficient and accurate in solving PDEs with the fractional Schr{o}dinger operator: $(-Delta)^s +|bs x|^{2mu}$ with $sin (0,1]$ and $mu>-1/2.$ Following the same spirit, we construct the second family of GHFs, dubbed as Muntz-type generalised Hermite functions (M-GHFs), which are orthogonal with respect to an inner product associated with the underlying Schr{o}dinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Muntz-type GHF spectral method leads to sparse matrices and spectrally accurate to some Schr{o}dinger eigenvalue problems.
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