LSFEM

It is known that the least-squares finite element method typically provides poor conservation in fluid motion because it minimizes the continuity equation in L²-norm instead of using exact discrete mass conservative form. One of the prototype domains considered was a tube of size 1x1x10. The standard first-order system least-square (FOSLS) finite element method in velocity/pressure Navier-Stokes first order formulation with piecewise quartic finite elements in coarse grid (mesh size h=1/4) yielded 47% of mass loss in this tube. However, by using FOSLL* with a modification of normal vectors in Newton iteration instead of FOSLS at the last iterate, mass loss was improved to 0.03%. For a given system of nonlinear partial differential equations L(U) = F, the linearization process (using Newton's method) yields a linear system LV=G and then FOSLS is used to minimize the linear system in L²-norm, that is, to find V such that V = arg min || LW – G ||. Typical Newton-FOSLS process repeats the above iteration. For a given linear system LV=G, the FOSLL* method solves the system, LL*V*=G, by minimizing the functional, ||L*V*-V||², with the dual variable, V*, and the L²-adjoint operator, L*, of L. Minimizing ||L*V*-V||² over V* in the domain of L* is accomplished by solving the weak problem of finding V* such that < L*V* , L*W > = < V , L*W > = < LV , W> = < G , W>, for every W in the domain of L*. Then, the solution we seek is V =L* V *. This equation shows that we can solve the dual problem with the given data (right-hand side) of the original problem without knowing the exact solution, V. As mentioned before FOSLL* was used to solve the linearized system at the last Newton iterate and mass loss was improved to 0.03%. While getting improved mass conservation by using FOSLL* only at the last stage of Newton iterations, questions arose what happens if FOSLL* was used in several stages in Newton iterations and how this process can be generalized. The main difficulties in applying FOSLL* in Newton iterations and the generalization is considered. We first observed the general ideas of Newton's method, FOSLS and FOSLL* approaches. Then we proposed difficulty which is arisen in the combination of Newton's iteration and FOSLL* then suggest several remedies for that. We have analyzed Newton-FOSLL* iteration for particular form of nonlinear partial differential equations.

Figure : Error with different weight parameters when the right hand side is f=(rho)^-3/2log(rho/1000)

• A least-squares finite element method for a nonlinear Stokes problem in glaciology

비선형 Stokes equations을 분석하고 negative-norm 최소자승유한요소법을 이용하여 그 근사해를 찾는 방법에 대해 연구 하였다. 최소화 문제를 정의하고 그에 대응되는 variational formulation을 정의 하였다. 적당한 Sobolev solution space를 정의하여 variational formulation의 해의 존재성과 유일성을 증명하였다. 다양한 시도를 통해 negative norm 수치 시뮬레이션을 완성하였고 2차 공간에서 성공적인 수치값을 얻어냈다.

A stationary Stokes problem with nonlinear rheology and with mixed no-slip and sliding basal boundary conditions is considered. The model describes the flow of ice in glaciers and ice sheets. A least-squares finite element method is developed and analyzed. The method does not require that the finite element spaces satisfy an inf-sup condition. Moreover, the usage of negative Sobolev norm in the least-squares functional allows for the use of standard piecewise polynomials spaces for both the velocity and pressure approximations. A Picard-type iterative method is used to linearize the Stokes problem.

• Maxwell's equations in three dimension with boundary singularities and discontinuous coefficients

FOSLL*를 이용한 비선형 편미분 방정식의 근사해 연구: 편미분 방정식의 근사 해를 찾는 방법 중 하나인 FOSLL* 방법은 least-squares(LS) 방법에 근간을 두고 있다. 통상적으로 방정식의 해는 계산공간과, 계수, 데이터 등의 적절한 smoothness 가정 하에서 Sobolev 공간에 속해 있다. 따라서 주어진 선형 편미분 방정식에서, 불연속 계수가 있든지, 계산공간에 singularity가 있으면 그 해 또한 singularity를 가지게 되고 이로 인해 LS방법뿐만 아니라 통상적인 유한요소법의 모든 방법의 이용이 어렵게 된다. 이를 극복하기 위해 FOSLL*방법이 개발 되었다. FOSLL* 방법은 그 dual 시스템을 고려하여 얻은 근사 해를 일종의 corrector역할을 하는 인자로써 이용하여 regularity가 낮은 근사 해를 찾아 주게 된다.

In the case that the domain has reentrant edges, the standard finite element method loses its global accuracy because of singularities on the boundary. To overcome this difficulty, FOSLL* is applied. FOSLL* is a methodology for solving PDEs using the dual operator. A modified FOSLL* method is developed that employs a partially weighted functional and allows the use of a standard finite element scheme without losing global accuracy.